diff --git a/Lib/functools.py b/Lib/functools.py
--- a/Lib/functools.py
+++ b/Lib/functools.py
@@ -55,23 +55,28 @@
     convert = {
         '__lt__': [('__gt__', lambda self, other: not (self < other or self == other)),
                    ('__le__', lambda self, other: self < other or self == other),
+                   ('__ne__', lambda self, other: not self == other),
                    ('__ge__', lambda self, other: not self < other)],
         '__le__': [('__ge__', lambda self, other: not self <= other or self == other),
                    ('__lt__', lambda self, other: self <= other and not self == other),
+                   ('__ne__', lambda self, other: not self == other),
                    ('__gt__', lambda self, other: not self <= other)],
         '__gt__': [('__lt__', lambda self, other: not (self > other or self == other)),
                    ('__ge__', lambda self, other: self > other or self == other),
+                   ('__ne__', lambda self, other: not self == other),
                    ('__le__', lambda self, other: not self > other)],
         '__ge__': [('__le__', lambda self, other: (not self >= other) or self == other),
                    ('__gt__', lambda self, other: self >= other and not self == other),
+                   ('__ne__', lambda self, other: not self == other),
                    ('__lt__', lambda self, other: not self >= other)]
     }
-    roots = set(dir(cls)) & set(convert)
+    defined_methods = set(dir(cls))
+    roots = defined_methods & set(convert)
     if not roots:
         raise ValueError('must define at least one ordering operation: < > <= >=')
     root = max(roots)       # prefer __lt__ to __le__ to __gt__ to __ge__
     for opname, opfunc in convert[root]:
-        if opname not in roots:
+        if opname not in defined_methods:
             opfunc.__name__ = opname
             opfunc.__doc__ = getattr(int, opname).__doc__
             setattr(cls, opname, opfunc)
diff --git a/Lib/test/test_functools.py b/Lib/test/test_functools.py
--- a/Lib/test/test_functools.py
+++ b/Lib/test/test_functools.py
@@ -474,6 +474,33 @@
         with self.assertRaises(TypeError):
             TestTO(8) <= ()
 
+    def test_bug_25732(self):
+        @functools.total_ordering
+        class A:
+            def __init__(self, value):
+                self.value = value
+            def __gt__(self, other):
+                return self.value > other.value
+            def __eq__(self, other):
+                return self.value == other.value
+        self.assertTrue(A(1) != A(2))
+        self.assertFalse(A(1) != A(1))
+
+        @functools.total_ordering
+        class A:
+            def __init__(self, value):
+                self.value = value
+            def __gt__(self, other):
+                return self.value > other.value
+            def __eq__(self, other):
+                return self.value == other.value
+            def __ne__(self, other):
+                raise RuntimeError(self, other)
+        with self.assertRaises(RuntimeError):
+            A(1) != A(2)
+        with self.assertRaises(RuntimeError):
+            A(1) != A(1)
+
 def test_main(verbose=None):
     test_classes = (
         TestPartial,