BUG: Fix (GH#61941) and unit test

Author: Aniketsy

pandas/_testing/asserters.py

@@ -323,7 +323,9 @@ def _check_types(left, right, obj: str = "Index") -> None:
     elif check_exact and check_categorical:
         if not left.equals(right):
             try:
-                mismatch = left._values != right._values
+                mismatch = ( 
+                    left._internal_get_values() != right._internal_get_values()
+                )
             except TypeError as e:
                 raise AssertionError(
                     f"{obj} cannot be compared due to incompatible"

pandas/tests/util/test_assert_index_equal.py

@@ -317,3 +317,11 @@ def test_assert_multi_index_dtype_check_categorical(check_categorical):
             tm.assert_index_equal(idx1, idx2, check_categorical=check_categorical)
     else:
         tm.assert_index_equal(idx1, idx2, check_categorical=check_categorical)
+
+def test_assert_index_equal_categorical_mismatch_categories():
+    # GH#61941
+    left = CategoricalIndex(["a", "b"], categories=["a", "b"])
+    right = CategoricalIndex(["a", "b"], categories=["b", "a"])
+
+    with pytest.raises(AssertionError, match="cannot be compared due to incompatible"):
+        tm.assert_index_equal(left, right, check_exact=True, check_categorical=True)