diff --git a/Doc/library/heapq.rst b/Doc/library/heapq.rst
index d3c4b920ba5..2bd0162a982 100644
--- a/Doc/library/heapq.rst
+++ b/Doc/library/heapq.rst
@@ -16,40 +16,56 @@
 This module provides an implementation of the heap queue algorithm, also known
 as the priority queue algorithm.
 
-Heaps are binary trees for which every parent node has a value less than or
-equal to any of its children.  We refer to this condition as the heap invariant.
+Min-heaps are binary trees for which every parent node has a value less than
+or equal to any of its children.
+We refer to this condition as the heap invariant.
 
-This implementation uses arrays for which
-``heap[k] <= heap[2*k+1]`` and ``heap[k] <= heap[2*k+2]`` for all *k*, counting
-elements from zero.  For the sake of comparison, non-existing elements are
-considered to be infinite.  The interesting property of a heap is that its
-smallest element is always the root, ``heap[0]``.
+For min-heaps, this implementation uses lists for which
+``heap[k] <= heap[2*k+1]`` and ``heap[k] <= heap[2*k+2]`` for all *k* for which
+the compared elements exist.  Elements are counted from zero.  The interesting
+property of a min-heap is that its smallest element is always the root,
+``heap[0]``.
 
-The API below differs from textbook heap algorithms in two aspects: (a) We use
-zero-based indexing.  This makes the relationship between the index for a node
-and the indexes for its children slightly less obvious, but is more suitable
-since Python uses zero-based indexing. (b) Our pop method returns the smallest
-item, not the largest (called a "min heap" in textbooks; a "max heap" is more
-common in texts because of its suitability for in-place sorting).
+Max-heaps satisfy the reverse invariant: every parent node has a value
+*greater* than any of its children.  These are implemented as lists for which
+``maxheap[2*k+1] <= maxheap[k]`` and ``maxheap[2*k+2] <= maxheap[k]`` for all
+*k* for which the compared elements exist.
+The root, ``maxheap[0]``, contains the *largest* element;
+``heap.sort(reverse=True)`` maintains the max-heap invariant.
 
-These two make it possible to view the heap as a regular Python list without
-surprises: ``heap[0]`` is the smallest item, and ``heap.sort()`` maintains the
-heap invariant!
+The :mod:`!heapq` API differs from textbook heap algorithms in two aspects: (a)
+We use zero-based indexing.  This makes the relationship between the index for
+a node and the indexes for its children slightly less obvious, but is more
+suitable since Python uses zero-based indexing. (b) Textbooks often focus on
+max-heaps, due to their suitability for in-place sorting. Our implementation
+favors min-heaps as they better correspond to Python :class:`lists <list>`.
 
-To create a heap, use a list initialized to ``[]``, or you can transform a
-populated list into a heap via function :func:`heapify`.
+These two aspects make it possible to view the heap as a regular Python list
+without surprises: ``heap[0]`` is the smallest item, and ``heap.sort()``
+maintains the heap invariant!
 
-The following functions are provided:
+Like :meth:`list.sort`, this implementation uses only the ``<`` operator
+for comparisons, for both min-heaps and max-heaps.
+
+In the API below, and in this documentation, the unqualified term *heap*
+generally refers to a min-heap.
+The API for max-heaps is named using a ``_max``  suffix.
+
+To create a heap, use a list initialized as ``[]``, or transform an existing list
+into a min-heap or max-heap using the :func:`heapify` or :func:`heapify_max`
+functions, respectively.
+
+The following functions are provided for min-heaps:
 
 
 .. function:: heappush(heap, item)
 
-   Push the value *item* onto the *heap*, maintaining the heap invariant.
+   Push the value *item* onto the *heap*, maintaining the min-heap invariant.
 
 
 .. function:: heappop(heap)
 
-   Pop and return the smallest item from the *heap*, maintaining the heap
+   Pop and return the smallest item from the *heap*, maintaining the min-heap
    invariant.  If the heap is empty, :exc:`IndexError` is raised.  To access the
    smallest item without popping it, use ``heap[0]``.
 
@@ -63,7 +79,7 @@ The following functions are provided:
 
 .. function:: heapify(x)
 
-   Transform list *x* into a heap, in-place, in linear time.
+   Transform list *x* into a min-heap, in-place, in linear time.
 
 
 .. function:: heapreplace(heap, item)
@@ -82,6 +98,56 @@ The following functions are provided:
    on the heap.
 
 
+For max-heaps, the following functions are provided:
+
+
+.. function:: heapify_max(x)
+
+   Transform list *x* into a max-heap, in-place, in linear time.
+
+   .. versionadded:: next
+
+
+.. function:: heappush_max(heap, item)
+
+   Push the value *item* onto the max-heap *heap*, maintaining the max-heap
+   invariant.
+
+   .. versionadded:: next
+
+
+.. function:: heappop_max(heap)
+
+   Pop and return the largest item from the max-heap *heap*, maintaining the
+   max-heap invariant.  If the max-heap is empty, :exc:`IndexError` is raised.
+   To access the largest item without popping it, use ``maxheap[0]``.
+
+   .. versionadded:: next
+
+
+.. function:: heappushpop_max(heap, item)
+
+   Push *item* on the max-heap *heap*, then pop and return the largest item
+   from *heap*.
+   The combined action runs more efficiently than :func:`heappush_max`
+   followed by a separate call to :func:`heappop_max`.
+
+   .. versionadded:: next
+
+
+.. function:: heapreplace_max(heap, item)
+
+   Pop and return the largest item from the max-heap *heap* and also push the
+   new *item*.
+   The max-heap size doesn't change. If the max-heap is empty,
+   :exc:`IndexError` is raised.
+
+   The value returned may be smaller than the *item* added.  Refer to the
+   analogous function :func:`heapreplace` for detailed usage notes.
+
+   .. versionadded:: next
+
+
 The module also offers three general purpose functions based on heaps.
 
 
diff --git a/Doc/whatsnew/3.14.rst b/Doc/whatsnew/3.14.rst
index f09e517d456..506a820175b 100644
--- a/Doc/whatsnew/3.14.rst
+++ b/Doc/whatsnew/3.14.rst
@@ -1114,6 +1114,18 @@ graphlib
   (Contributed by Daniel Pope in :gh:`130914`)
 
 
+heapq
+-----
+
+* Add functions for working with max-heaps:
+
+  * :func:`heapq.heapify_max`,
+  * :func:`heapq.heappush_max`,
+  * :func:`heapq.heappop_max`,
+  * :func:`heapq.heapreplace_max`
+  * :func:`heapq.heappushpop_max`
+
+
 hmac
 ----
 
diff --git a/Lib/heapq.py b/Lib/heapq.py
index 9649da251f2..6ceb211f1ca 100644
--- a/Lib/heapq.py
+++ b/Lib/heapq.py
@@ -178,7 +178,7 @@ def heapify(x):
     for i in reversed(range(n//2)):
         _siftup(x, i)
 
-def _heappop_max(heap):
+def heappop_max(heap):
     """Maxheap version of a heappop."""
     lastelt = heap.pop()    # raises appropriate IndexError if heap is empty
     if heap:
@@ -188,19 +188,32 @@ def _heappop_max(heap):
         return returnitem
     return lastelt
 
-def _heapreplace_max(heap, item):
+def heapreplace_max(heap, item):
     """Maxheap version of a heappop followed by a heappush."""
     returnitem = heap[0]    # raises appropriate IndexError if heap is empty
     heap[0] = item
     _siftup_max(heap, 0)
     return returnitem
 
-def _heapify_max(x):
+def heappush_max(heap, item):
+    """Maxheap version of a heappush."""
+    heap.append(item)
+    _siftdown_max(heap, 0, len(heap)-1)
+
+def heappushpop_max(heap, item):
+    """Maxheap fast version of a heappush followed by a heappop."""
+    if heap and item < heap[0]:
+        item, heap[0] = heap[0], item
+        _siftup_max(heap, 0)
+    return item
+
+def heapify_max(x):
     """Transform list into a maxheap, in-place, in O(len(x)) time."""
     n = len(x)
     for i in reversed(range(n//2)):
         _siftup_max(x, i)
 
+
 # 'heap' is a heap at all indices >= startpos, except possibly for pos.  pos
 # is the index of a leaf with a possibly out-of-order value.  Restore the
 # heap invariant.
@@ -335,9 +348,9 @@ def merge(*iterables, key=None, reverse=False):
     h_append = h.append
 
     if reverse:
-        _heapify = _heapify_max
-        _heappop = _heappop_max
-        _heapreplace = _heapreplace_max
+        _heapify = heapify_max
+        _heappop = heappop_max
+        _heapreplace = heapreplace_max
         direction = -1
     else:
         _heapify = heapify
@@ -490,10 +503,10 @@ def nsmallest(n, iterable, key=None):
         result = [(elem, i) for i, elem in zip(range(n), it)]
         if not result:
             return result
-        _heapify_max(result)
+        heapify_max(result)
         top = result[0][0]
         order = n
-        _heapreplace = _heapreplace_max
+        _heapreplace = heapreplace_max
         for elem in it:
             if elem < top:
                 _heapreplace(result, (elem, order))
@@ -507,10 +520,10 @@ def nsmallest(n, iterable, key=None):
     result = [(key(elem), i, elem) for i, elem in zip(range(n), it)]
     if not result:
         return result
-    _heapify_max(result)
+    heapify_max(result)
     top = result[0][0]
     order = n
-    _heapreplace = _heapreplace_max
+    _heapreplace = heapreplace_max
     for elem in it:
         k = key(elem)
         if k < top:
@@ -583,19 +596,13 @@ try:
     from _heapq import *
 except ImportError:
     pass
-try:
-    from _heapq import _heapreplace_max
-except ImportError:
-    pass
-try:
-    from _heapq import _heapify_max
-except ImportError:
-    pass
-try:
-    from _heapq import _heappop_max
-except ImportError:
-    pass
 
+# For backwards compatibility
+_heappop_max  = heappop_max
+_heapreplace_max = heapreplace_max
+_heappush_max = heappush_max
+_heappushpop_max = heappushpop_max
+_heapify_max = heapify_max
 
 if __name__ == "__main__":
 
diff --git a/Lib/test/test_heapq.py b/Lib/test/test_heapq.py
index 1aa8e4e2897..d6623fee9bb 100644
--- a/Lib/test/test_heapq.py
+++ b/Lib/test/test_heapq.py
@@ -13,8 +13,9 @@ c_heapq = import_helper.import_fresh_module('heapq', fresh=['_heapq'])
 
 # _heapq.nlargest/nsmallest are saved in heapq._nlargest/_smallest when
 # _heapq is imported, so check them there
-func_names = ['heapify', 'heappop', 'heappush', 'heappushpop', 'heapreplace',
-              '_heappop_max', '_heapreplace_max', '_heapify_max']
+func_names = ['heapify', 'heappop', 'heappush', 'heappushpop', 'heapreplace']
+# Add max-heap variants
+func_names += [func + '_max' for func in func_names]
 
 class TestModules(TestCase):
     def test_py_functions(self):
@@ -24,7 +25,7 @@ class TestModules(TestCase):
     @skipUnless(c_heapq, 'requires _heapq')
     def test_c_functions(self):
         for fname in func_names:
-            self.assertEqual(getattr(c_heapq, fname).__module__, '_heapq')
+            self.assertEqual(getattr(c_heapq, fname).__module__, '_heapq', fname)
 
 
 def load_tests(loader, tests, ignore):
@@ -74,6 +75,34 @@ class TestHeap:
         except AttributeError:
             pass
 
+    def test_max_push_pop(self):
+        # 1) Push 256 random numbers and pop them off, verifying all's OK.
+        heap = []
+        data = []
+        self.check_max_invariant(heap)
+        for i in range(256):
+            item = random.random()
+            data.append(item)
+            self.module.heappush_max(heap, item)
+            self.check_max_invariant(heap)
+        results = []
+        while heap:
+            item = self.module.heappop_max(heap)
+            self.check_max_invariant(heap)
+            results.append(item)
+        data_sorted = data[:]
+        data_sorted.sort(reverse=True)
+
+        self.assertEqual(data_sorted, results)
+        # 2) Check that the invariant holds for a sorted array
+        self.check_max_invariant(results)
+
+        self.assertRaises(TypeError, self.module.heappush_max, [])
+
+        exc_types = (AttributeError, TypeError)
+        self.assertRaises(exc_types, self.module.heappush_max, None, None)
+        self.assertRaises(exc_types, self.module.heappop_max, None)
+
     def check_invariant(self, heap):
         # Check the heap invariant.
         for pos, item in enumerate(heap):
@@ -81,6 +110,11 @@ class TestHeap:
                 parentpos = (pos-1) >> 1
                 self.assertTrue(heap[parentpos] <= item)
 
+    def check_max_invariant(self, heap):
+        for pos, item in enumerate(heap[1:], start=1):
+            parentpos = (pos - 1) >> 1
+            self.assertGreaterEqual(heap[parentpos], item)
+
     def test_heapify(self):
         for size in list(range(30)) + [20000]:
             heap = [random.random() for dummy in range(size)]
@@ -89,6 +123,14 @@ class TestHeap:
 
         self.assertRaises(TypeError, self.module.heapify, None)
 
+    def test_heapify_max(self):
+        for size in list(range(30)) + [20000]:
+            heap = [random.random() for dummy in range(size)]
+            self.module.heapify_max(heap)
+            self.check_max_invariant(heap)
+
+        self.assertRaises(TypeError, self.module.heapify_max, None)
+
     def test_naive_nbest(self):
         data = [random.randrange(2000) for i in range(1000)]
         heap = []
@@ -109,10 +151,7 @@ class TestHeap:
 
     def test_nbest(self):
         # Less-naive "N-best" algorithm, much faster (if len(data) is big
-        # enough <wink>) than sorting all of data.  However, if we had a max
-        # heap instead of a min heap, it could go faster still via
-        # heapify'ing all of data (linear time), then doing 10 heappops
-        # (10 log-time steps).
+        # enough <wink>) than sorting all of data.
         data = [random.randrange(2000) for i in range(1000)]
         heap = data[:10]
         self.module.heapify(heap)
@@ -125,6 +164,17 @@ class TestHeap:
         self.assertRaises(TypeError, self.module.heapreplace, None, None)
         self.assertRaises(IndexError, self.module.heapreplace, [], None)
 
+    def test_nbest_maxheap(self):
+        # With a max heap instead of a min heap, the "N-best" algorithm can
+        # go even faster still via heapify'ing all of data (linear time), then
+        # doing 10 heappops (10 log-time steps).
+        data = [random.randrange(2000) for i in range(1000)]
+        heap = data[:]
+        self.module.heapify_max(heap)
+        result = [self.module.heappop_max(heap) for _ in range(10)]
+        result.reverse()
+        self.assertEqual(result, sorted(data)[-10:])
+
     def test_nbest_with_pushpop(self):
         data = [random.randrange(2000) for i in range(1000)]
         heap = data[:10]
@@ -134,6 +184,62 @@ class TestHeap:
         self.assertEqual(list(self.heapiter(heap)), sorted(data)[-10:])
         self.assertEqual(self.module.heappushpop([], 'x'), 'x')
 
+    def test_naive_nworst(self):
+        # Max-heap variant of "test_naive_nbest"
+        data = [random.randrange(2000) for i in range(1000)]
+        heap = []
+        for item in data:
+            self.module.heappush_max(heap, item)
+            if len(heap) > 10:
+                self.module.heappop_max(heap)
+        heap.sort()
+        expected = sorted(data)[:10]
+        self.assertEqual(heap, expected)
+
+    def heapiter_max(self, heap):
+        # An iterator returning a max-heap's elements, largest-first.
+        try:
+            while 1:
+                yield self.module.heappop_max(heap)
+        except IndexError:
+            pass
+
+    def test_nworst(self):
+        # Max-heap variant of "test_nbest"
+        data = [random.randrange(2000) for i in range(1000)]
+        heap = data[:10]
+        self.module.heapify_max(heap)
+        for item in data[10:]:
+            if item < heap[0]:  # this gets rarer the longer we run
+                self.module.heapreplace_max(heap, item)
+        expected = sorted(data, reverse=True)[-10:]
+        self.assertEqual(list(self.heapiter_max(heap)), expected)
+
+        self.assertRaises(TypeError, self.module.heapreplace_max, None)
+        self.assertRaises(TypeError, self.module.heapreplace_max, None, None)
+        self.assertRaises(IndexError, self.module.heapreplace_max, [], None)
+
+    def test_nworst_minheap(self):
+        # Min-heap variant of "test_nbest_maxheap"
+        data = [random.randrange(2000) for i in range(1000)]
+        heap = data[:]
+        self.module.heapify(heap)
+        result = [self.module.heappop(heap) for _ in range(10)]
+        result.reverse()
+        expected = sorted(data, reverse=True)[-10:]
+        self.assertEqual(result, expected)
+
+    def test_nworst_with_pushpop(self):
+        # Max-heap variant of "test_nbest_with_pushpop"
+        data = [random.randrange(2000) for i in range(1000)]
+        heap = data[:10]
+        self.module.heapify_max(heap)
+        for item in data[10:]:
+            self.module.heappushpop_max(heap, item)
+        expected = sorted(data, reverse=True)[-10:]
+        self.assertEqual(list(self.heapiter_max(heap)), expected)
+        self.assertEqual(self.module.heappushpop_max([], 'x'), 'x')
+
     def test_heappushpop(self):
         h = []
         x = self.module.heappushpop(h, 10)
@@ -153,12 +259,31 @@ class TestHeap:
         x = self.module.heappushpop(h, 11)
         self.assertEqual((h, x), ([11], 10))
 
+    def test_heappushpop_max(self):
+        h = []
+        x = self.module.heappushpop_max(h, 10)
+        self.assertTupleEqual((h, x), ([], 10))
+
+        h = [10]
+        x = self.module.heappushpop_max(h, 10.0)
+        self.assertTupleEqual((h, x), ([10], 10.0))
+        self.assertIsInstance(h[0], int)
+        self.assertIsInstance(x, float)
+
+        h = [10]
+        x = self.module.heappushpop_max(h, 11)
+        self.assertTupleEqual((h, x), ([10], 11))
+
+        h = [10]
+        x = self.module.heappushpop_max(h, 9)
+        self.assertTupleEqual((h, x), ([9], 10))
+
     def test_heappop_max(self):
-        # _heapop_max has an optimization for one-item lists which isn't
+        # heapop_max has an optimization for one-item lists which isn't
         # covered in other tests, so test that case explicitly here
         h = [3, 2]
-        self.assertEqual(self.module._heappop_max(h), 3)
-        self.assertEqual(self.module._heappop_max(h), 2)
+        self.assertEqual(self.module.heappop_max(h), 3)
+        self.assertEqual(self.module.heappop_max(h), 2)
 
     def test_heapsort(self):
         # Exercise everything with repeated heapsort checks
@@ -175,6 +300,20 @@ class TestHeap:
             heap_sorted = [self.module.heappop(heap) for i in range(size)]
             self.assertEqual(heap_sorted, sorted(data))
 
+    def test_heapsort_max(self):
+        for trial in range(100):
+            size = random.randrange(50)
+            data = [random.randrange(25) for i in range(size)]
+            if trial & 1:     # Half of the time, use heapify_max
+                heap = data[:]
+                self.module.heapify_max(heap)
+            else:             # The rest of the time, use heappush_max
+                heap = []
+                for item in data:
+                    self.module.heappush_max(heap, item)
+            heap_sorted = [self.module.heappop_max(heap) for i in range(size)]
+            self.assertEqual(heap_sorted, sorted(data, reverse=True))
+
     def test_merge(self):
         inputs = []
         for i in range(random.randrange(25)):
@@ -377,16 +516,20 @@ class SideEffectLT:
 class TestErrorHandling:
 
     def test_non_sequence(self):
-        for f in (self.module.heapify, self.module.heappop):
+        for f in (self.module.heapify, self.module.heappop,
+                  self.module.heapify_max, self.module.heappop_max):
             self.assertRaises((TypeError, AttributeError), f, 10)
         for f in (self.module.heappush, self.module.heapreplace,
+                  self.module.heappush_max, self.module.heapreplace_max,
                   self.module.nlargest, self.module.nsmallest):
             self.assertRaises((TypeError, AttributeError), f, 10, 10)
 
     def test_len_only(self):
-        for f in (self.module.heapify, self.module.heappop):
+        for f in (self.module.heapify, self.module.heappop,
+                  self.module.heapify_max, self.module.heappop_max):
             self.assertRaises((TypeError, AttributeError), f, LenOnly())
-        for f in (self.module.heappush, self.module.heapreplace):
+        for f in (self.module.heappush, self.module.heapreplace,
+                  self.module.heappush_max, self.module.heapreplace_max):
             self.assertRaises((TypeError, AttributeError), f, LenOnly(), 10)
         for f in (self.module.nlargest, self.module.nsmallest):
             self.assertRaises(TypeError, f, 2, LenOnly())
@@ -395,7 +538,8 @@ class TestErrorHandling:
         seq = [CmpErr(), CmpErr(), CmpErr()]
         for f in (self.module.heapify, self.module.heappop):
             self.assertRaises(ZeroDivisionError, f, seq)
-        for f in (self.module.heappush, self.module.heapreplace):
+        for f in (self.module.heappush, self.module.heapreplace,
+                  self.module.heappush_max, self.module.heapreplace_max):
             self.assertRaises(ZeroDivisionError, f, seq, 10)
         for f in (self.module.nlargest, self.module.nsmallest):
             self.assertRaises(ZeroDivisionError, f, 2, seq)
@@ -403,6 +547,8 @@ class TestErrorHandling:
     def test_arg_parsing(self):
         for f in (self.module.heapify, self.module.heappop,
                   self.module.heappush, self.module.heapreplace,
+                  self.module.heapify_max, self.module.heappop_max,
+                  self.module.heappush_max, self.module.heapreplace_max,
                   self.module.nlargest, self.module.nsmallest):
             self.assertRaises((TypeError, AttributeError), f, 10)
 
@@ -424,6 +570,10 @@ class TestErrorHandling:
         # Python version raises IndexError, C version RuntimeError
         with self.assertRaises((IndexError, RuntimeError)):
             self.module.heappush(heap, SideEffectLT(5, heap))
+        heap = []
+        heap.extend(SideEffectLT(i, heap) for i in range(200))
+        with self.assertRaises((IndexError, RuntimeError)):
+            self.module.heappush_max(heap, SideEffectLT(5, heap))
 
     def test_heappop_mutating_heap(self):
         heap = []
@@ -431,8 +581,12 @@ class TestErrorHandling:
         # Python version raises IndexError, C version RuntimeError
         with self.assertRaises((IndexError, RuntimeError)):
             self.module.heappop(heap)
+        heap = []
+        heap.extend(SideEffectLT(i, heap) for i in range(200))
+        with self.assertRaises((IndexError, RuntimeError)):
+            self.module.heappop_max(heap)
 
-    def test_comparison_operator_modifiying_heap(self):
+    def test_comparison_operator_modifying_heap(self):
         # See bpo-39421: Strong references need to be taken
         # when comparing objects as they can alter the heap
         class EvilClass(int):
@@ -444,7 +598,7 @@ class TestErrorHandling:
         self.module.heappush(heap, EvilClass(0))
         self.assertRaises(IndexError, self.module.heappushpop, heap, 1)
 
-    def test_comparison_operator_modifiying_heap_two_heaps(self):
+    def test_comparison_operator_modifying_heap_two_heaps(self):
 
         class h(int):
             def __lt__(self, o):
@@ -464,6 +618,17 @@ class TestErrorHandling:
         self.assertRaises((IndexError, RuntimeError), self.module.heappush, list1, g(1))
         self.assertRaises((IndexError, RuntimeError), self.module.heappush, list2, h(1))
 
+        list1, list2 = [], []
+
+        self.module.heappush_max(list1, h(0))
+        self.module.heappush_max(list2, g(0))
+        self.module.heappush_max(list1, g(1))
+        self.module.heappush_max(list2, h(1))
+
+        self.assertRaises((IndexError, RuntimeError), self.module.heappush_max, list1, g(1))
+        self.assertRaises((IndexError, RuntimeError), self.module.heappush_max, list2, h(1))
+
+
 class TestErrorHandlingPython(TestErrorHandling, TestCase):
     module = py_heapq
 
diff --git a/Misc/NEWS.d/next/Library/2025-03-01-15-00-00.gh-issue-110067.1ad3as.rst b/Misc/NEWS.d/next/Library/2025-03-01-15-00-00.gh-issue-110067.1ad3as.rst
new file mode 100644
index 00000000000..98e125f4966
--- /dev/null
+++ b/Misc/NEWS.d/next/Library/2025-03-01-15-00-00.gh-issue-110067.1ad3as.rst
@@ -0,0 +1,5 @@
+Make :mod:`heapq` max-heap functions :func:`heapq.heapify_max`, :func:`heapq.heappush_max`,
+:func:`heapq.heappop_max`, and :func:`heapq.heapreplace_max` public.
+Previous underscored naming is kept for backwards compatibility.
+Additionally, the missing function :func:`heapq.heappushpop_max` has been added
+to both the C and Python implementations.
diff --git a/Modules/_heapqmodule.c b/Modules/_heapqmodule.c
index 80fe9cff985..095866eec7d 100644
--- a/Modules/_heapqmodule.c
+++ b/Modules/_heapqmodule.c
@@ -480,9 +480,33 @@ siftup_max(PyListObject *heap, Py_ssize_t pos)
     return siftdown_max(heap, startpos, pos);
 }
 
+/*[clinic input]
+_heapq.heappush_max
+
+    heap: object(subclass_of='&PyList_Type')
+    item: object
+    /
+
+Push item onto max heap, maintaining the heap invariant.
+[clinic start generated code]*/
+
+static PyObject *
+_heapq_heappush_max_impl(PyObject *module, PyObject *heap, PyObject *item)
+/*[clinic end generated code: output=c869d5f9deb08277 input=4743d7db137b6e2b]*/
+{
+    if (PyList_Append(heap, item)) {
+        return NULL;
+    }
+
+    if (siftdown_max((PyListObject *)heap, 0, PyList_GET_SIZE(heap)-1)) {
+        return NULL;
+    }
+
+    Py_RETURN_NONE;
+}
 
 /*[clinic input]
-_heapq._heappop_max
+_heapq.heappop_max
 
     heap: object(subclass_of='&PyList_Type')
     /
@@ -491,14 +515,14 @@ Maxheap variant of heappop.
 [clinic start generated code]*/
 
 static PyObject *
-_heapq__heappop_max_impl(PyObject *module, PyObject *heap)
-/*[clinic end generated code: output=9e77aadd4e6a8760 input=362c06e1c7484793]*/
+_heapq_heappop_max_impl(PyObject *module, PyObject *heap)
+/*[clinic end generated code: output=2f051195ab404b77 input=e62b14016a5a26de]*/
 {
     return heappop_internal(heap, siftup_max);
 }
 
 /*[clinic input]
-_heapq._heapreplace_max
+_heapq.heapreplace_max
 
     heap: object(subclass_of='&PyList_Type')
     item: object
@@ -508,15 +532,14 @@ Maxheap variant of heapreplace.
 [clinic start generated code]*/
 
 static PyObject *
-_heapq__heapreplace_max_impl(PyObject *module, PyObject *heap,
-                             PyObject *item)
-/*[clinic end generated code: output=8ad7545e4a5e8adb input=f2dd27cbadb948d7]*/
+_heapq_heapreplace_max_impl(PyObject *module, PyObject *heap, PyObject *item)
+/*[clinic end generated code: output=8770778b5a9cbe9b input=21a3d28d757c881c]*/
 {
     return heapreplace_internal(heap, item, siftup_max);
 }
 
 /*[clinic input]
-_heapq._heapify_max
+_heapq.heapify_max
 
     heap: object(subclass_of='&PyList_Type')
     /
@@ -525,21 +548,74 @@ Maxheap variant of heapify.
 [clinic start generated code]*/
 
 static PyObject *
-_heapq__heapify_max_impl(PyObject *module, PyObject *heap)
-/*[clinic end generated code: output=2cb028beb4a8b65e input=c1f765ee69f124b8]*/
+_heapq_heapify_max_impl(PyObject *module, PyObject *heap)
+/*[clinic end generated code: output=8401af3856529807 input=edda4255728c431e]*/
 {
     return heapify_internal(heap, siftup_max);
 }
 
+/*[clinic input]
+_heapq.heappushpop_max
+
+    heap: object(subclass_of='&PyList_Type')
+    item: object
+    /
+
+Maxheap variant of heappushpop.
+
+The combined action runs more efficiently than heappush_max() followed by
+a separate call to heappop_max().
+[clinic start generated code]*/
+
+static PyObject *
+_heapq_heappushpop_max_impl(PyObject *module, PyObject *heap, PyObject *item)
+/*[clinic end generated code: output=ff0019f0941aca0d input=525a843013cbd6c0]*/
+{
+    PyObject *returnitem;
+    int cmp;
+
+    if (PyList_GET_SIZE(heap) == 0) {
+        return Py_NewRef(item);
+    }
+
+    PyObject *top = PyList_GET_ITEM(heap, 0);
+    Py_INCREF(top);
+    cmp = PyObject_RichCompareBool(item, top, Py_LT);
+    Py_DECREF(top);
+    if (cmp < 0) {
+        return NULL;
+    }
+    if (cmp == 0) {
+        return Py_NewRef(item);
+    }
+
+    if (PyList_GET_SIZE(heap) == 0) {
+        PyErr_SetString(PyExc_IndexError, "index out of range");
+        return NULL;
+    }
+
+    returnitem = PyList_GET_ITEM(heap, 0);
+    PyList_SET_ITEM(heap, 0, Py_NewRef(item));
+    if (siftup_max((PyListObject *)heap, 0) < 0) {
+        Py_DECREF(returnitem);
+        return NULL;
+    }
+    return returnitem;
+}
+
 static PyMethodDef heapq_methods[] = {
     _HEAPQ_HEAPPUSH_METHODDEF
     _HEAPQ_HEAPPUSHPOP_METHODDEF
     _HEAPQ_HEAPPOP_METHODDEF
     _HEAPQ_HEAPREPLACE_METHODDEF
     _HEAPQ_HEAPIFY_METHODDEF
-    _HEAPQ__HEAPPOP_MAX_METHODDEF
-    _HEAPQ__HEAPIFY_MAX_METHODDEF
-    _HEAPQ__HEAPREPLACE_MAX_METHODDEF
+
+    _HEAPQ_HEAPPUSH_MAX_METHODDEF
+    _HEAPQ_HEAPPUSHPOP_MAX_METHODDEF
+    _HEAPQ_HEAPPOP_MAX_METHODDEF
+    _HEAPQ_HEAPREPLACE_MAX_METHODDEF
+    _HEAPQ_HEAPIFY_MAX_METHODDEF
+
     {NULL, NULL}           /* sentinel */
 };
 
diff --git a/Modules/clinic/_heapqmodule.c.h b/Modules/clinic/_heapqmodule.c.h
index 90463079907..81d10862726 100644
--- a/Modules/clinic/_heapqmodule.c.h
+++ b/Modules/clinic/_heapqmodule.c.h
@@ -175,96 +175,166 @@ exit:
     return return_value;
 }
 
-PyDoc_STRVAR(_heapq__heappop_max__doc__,
-"_heappop_max($module, heap, /)\n"
+PyDoc_STRVAR(_heapq_heappush_max__doc__,
+"heappush_max($module, heap, item, /)\n"
 "--\n"
 "\n"
-"Maxheap variant of heappop.");
+"Push item onto max heap, maintaining the heap invariant.");
 
-#define _HEAPQ__HEAPPOP_MAX_METHODDEF    \
-    {"_heappop_max", (PyCFunction)_heapq__heappop_max, METH_O, _heapq__heappop_max__doc__},
+#define _HEAPQ_HEAPPUSH_MAX_METHODDEF    \
+    {"heappush_max", _PyCFunction_CAST(_heapq_heappush_max), METH_FASTCALL, _heapq_heappush_max__doc__},
 
 static PyObject *
-_heapq__heappop_max_impl(PyObject *module, PyObject *heap);
+_heapq_heappush_max_impl(PyObject *module, PyObject *heap, PyObject *item);
 
 static PyObject *
-_heapq__heappop_max(PyObject *module, PyObject *arg)
-{
-    PyObject *return_value = NULL;
-    PyObject *heap;
-
-    if (!PyList_Check(arg)) {
-        _PyArg_BadArgument("_heappop_max", "argument", "list", arg);
-        goto exit;
-    }
-    heap = arg;
-    return_value = _heapq__heappop_max_impl(module, heap);
-
-exit:
-    return return_value;
-}
-
-PyDoc_STRVAR(_heapq__heapreplace_max__doc__,
-"_heapreplace_max($module, heap, item, /)\n"
-"--\n"
-"\n"
-"Maxheap variant of heapreplace.");
-
-#define _HEAPQ__HEAPREPLACE_MAX_METHODDEF    \
-    {"_heapreplace_max", _PyCFunction_CAST(_heapq__heapreplace_max), METH_FASTCALL, _heapq__heapreplace_max__doc__},
-
-static PyObject *
-_heapq__heapreplace_max_impl(PyObject *module, PyObject *heap,
-                             PyObject *item);
-
-static PyObject *
-_heapq__heapreplace_max(PyObject *module, PyObject *const *args, Py_ssize_t nargs)
+_heapq_heappush_max(PyObject *module, PyObject *const *args, Py_ssize_t nargs)
 {
     PyObject *return_value = NULL;
     PyObject *heap;
     PyObject *item;
 
-    if (!_PyArg_CheckPositional("_heapreplace_max", nargs, 2, 2)) {
+    if (!_PyArg_CheckPositional("heappush_max", nargs, 2, 2)) {
         goto exit;
     }
     if (!PyList_Check(args[0])) {
-        _PyArg_BadArgument("_heapreplace_max", "argument 1", "list", args[0]);
+        _PyArg_BadArgument("heappush_max", "argument 1", "list", args[0]);
         goto exit;
     }
     heap = args[0];
     item = args[1];
-    return_value = _heapq__heapreplace_max_impl(module, heap, item);
+    return_value = _heapq_heappush_max_impl(module, heap, item);
 
 exit:
     return return_value;
 }
 
-PyDoc_STRVAR(_heapq__heapify_max__doc__,
-"_heapify_max($module, heap, /)\n"
+PyDoc_STRVAR(_heapq_heappop_max__doc__,
+"heappop_max($module, heap, /)\n"
 "--\n"
 "\n"
-"Maxheap variant of heapify.");
+"Maxheap variant of heappop.");
 
-#define _HEAPQ__HEAPIFY_MAX_METHODDEF    \
-    {"_heapify_max", (PyCFunction)_heapq__heapify_max, METH_O, _heapq__heapify_max__doc__},
+#define _HEAPQ_HEAPPOP_MAX_METHODDEF    \
+    {"heappop_max", (PyCFunction)_heapq_heappop_max, METH_O, _heapq_heappop_max__doc__},
 
 static PyObject *
-_heapq__heapify_max_impl(PyObject *module, PyObject *heap);
+_heapq_heappop_max_impl(PyObject *module, PyObject *heap);
 
 static PyObject *
-_heapq__heapify_max(PyObject *module, PyObject *arg)
+_heapq_heappop_max(PyObject *module, PyObject *arg)
 {
     PyObject *return_value = NULL;
     PyObject *heap;
 
     if (!PyList_Check(arg)) {
-        _PyArg_BadArgument("_heapify_max", "argument", "list", arg);
+        _PyArg_BadArgument("heappop_max", "argument", "list", arg);
         goto exit;
     }
     heap = arg;
-    return_value = _heapq__heapify_max_impl(module, heap);
+    return_value = _heapq_heappop_max_impl(module, heap);
 
 exit:
     return return_value;
 }
-/*[clinic end generated code: output=05f2afdf3bc54c9d input=a9049054013a1b77]*/
+
+PyDoc_STRVAR(_heapq_heapreplace_max__doc__,
+"heapreplace_max($module, heap, item, /)\n"
+"--\n"
+"\n"
+"Maxheap variant of heapreplace.");
+
+#define _HEAPQ_HEAPREPLACE_MAX_METHODDEF    \
+    {"heapreplace_max", _PyCFunction_CAST(_heapq_heapreplace_max), METH_FASTCALL, _heapq_heapreplace_max__doc__},
+
+static PyObject *
+_heapq_heapreplace_max_impl(PyObject *module, PyObject *heap, PyObject *item);
+
+static PyObject *
+_heapq_heapreplace_max(PyObject *module, PyObject *const *args, Py_ssize_t nargs)
+{
+    PyObject *return_value = NULL;
+    PyObject *heap;
+    PyObject *item;
+
+    if (!_PyArg_CheckPositional("heapreplace_max", nargs, 2, 2)) {
+        goto exit;
+    }
+    if (!PyList_Check(args[0])) {
+        _PyArg_BadArgument("heapreplace_max", "argument 1", "list", args[0]);
+        goto exit;
+    }
+    heap = args[0];
+    item = args[1];
+    return_value = _heapq_heapreplace_max_impl(module, heap, item);
+
+exit:
+    return return_value;
+}
+
+PyDoc_STRVAR(_heapq_heapify_max__doc__,
+"heapify_max($module, heap, /)\n"
+"--\n"
+"\n"
+"Maxheap variant of heapify.");
+
+#define _HEAPQ_HEAPIFY_MAX_METHODDEF    \
+    {"heapify_max", (PyCFunction)_heapq_heapify_max, METH_O, _heapq_heapify_max__doc__},
+
+static PyObject *
+_heapq_heapify_max_impl(PyObject *module, PyObject *heap);
+
+static PyObject *
+_heapq_heapify_max(PyObject *module, PyObject *arg)
+{
+    PyObject *return_value = NULL;
+    PyObject *heap;
+
+    if (!PyList_Check(arg)) {
+        _PyArg_BadArgument("heapify_max", "argument", "list", arg);
+        goto exit;
+    }
+    heap = arg;
+    return_value = _heapq_heapify_max_impl(module, heap);
+
+exit:
+    return return_value;
+}
+
+PyDoc_STRVAR(_heapq_heappushpop_max__doc__,
+"heappushpop_max($module, heap, item, /)\n"
+"--\n"
+"\n"
+"Maxheap variant of heappushpop.\n"
+"\n"
+"The combined action runs more efficiently than heappush_max() followed by\n"
+"a separate call to heappop_max().");
+
+#define _HEAPQ_HEAPPUSHPOP_MAX_METHODDEF    \
+    {"heappushpop_max", _PyCFunction_CAST(_heapq_heappushpop_max), METH_FASTCALL, _heapq_heappushpop_max__doc__},
+
+static PyObject *
+_heapq_heappushpop_max_impl(PyObject *module, PyObject *heap, PyObject *item);
+
+static PyObject *
+_heapq_heappushpop_max(PyObject *module, PyObject *const *args, Py_ssize_t nargs)
+{
+    PyObject *return_value = NULL;
+    PyObject *heap;
+    PyObject *item;
+
+    if (!_PyArg_CheckPositional("heappushpop_max", nargs, 2, 2)) {
+        goto exit;
+    }
+    if (!PyList_Check(args[0])) {
+        _PyArg_BadArgument("heappushpop_max", "argument 1", "list", args[0]);
+        goto exit;
+    }
+    heap = args[0];
+    item = args[1];
+    return_value = _heapq_heappushpop_max_impl(module, heap, item);
+
+exit:
+    return return_value;
+}
+/*[clinic end generated code: output=f55d8595ce150c76 input=a9049054013a1b77]*/