BUG: Categorical comparison with unordered (pandas-dev#16339)

Fixes categorical comparison operations improperly considering
ordering when two unordered categoricals are compared.

Closes pandas-dev#16014

doc/source/categorical.rst

@@ -453,6 +453,14 @@ the original values:
 
     np.asarray(cat) > base
 
+When you compare two unordered categoricals with the same categories, the order is not considered:
+
+.. ipython:: python
+
+   c1 = pd.Categorical(['a', 'b'], categories=['a', 'b'], ordered=False)
+   c2 = pd.Categorical(['a', 'b'], categories=['b', 'a'], ordered=False)
+   c1 == c2
+
 Operations
 ----------
 

doc/source/whatsnew/v0.20.2.txt

@@ -86,7 +86,10 @@ Numeric
 ^^^^^^^
 
 
+Categorical
+^^^^^^^^^^^
 
+- Fixed comparison operations considering the order of the categories when both categoricals are unordered (:issue:`16014`)
 
 Other
 ^^^^^

pandas/core/categorical.py

@@ -55,17 +55,31 @@ def f(self, other):
                                 "equality or not")
         if isinstance(other, Categorical):
             # Two Categoricals can only be be compared if the categories are
-            # the same
-            if ((len(self.categories) != len(other.categories)) or
-                    not ((self.categories == other.categories).all())):
-                raise TypeError("Categoricals can only be compared if "
-                                "'categories' are the same")
+            # the same (maybe up to ordering, depending on ordered)
+
+            msg = ("Categoricals can only be compared if "
+                   "'categories' are the same.")
+            if len(self.categories) != len(other.categories):
+                raise TypeError(msg + " Categories are different lengths")
+            elif (self.ordered and not (self.categories ==
+                                        other.categories).all()):
+                raise TypeError(msg)
+            elif not set(self.categories) == set(other.categories):
+                raise TypeError(msg)
+
             if not (self.ordered == other.ordered):
                 raise TypeError("Categoricals can only be compared if "
                                 "'ordered' is the same")
-            na_mask = (self._codes == -1) | (other._codes == -1)
+            if not self.ordered and not self.categories.equals(
+                    other.categories):
+                # both unordered and different order
+                other_codes = _get_codes_for_values(other, self.categories)
+            else:
+                other_codes = other._codes
+
+            na_mask = (self._codes == -1) | (other_codes == -1)
             f = getattr(self._codes, op)
-            ret = f(other._codes)
+            ret = f(other_codes)
             if na_mask.any():
                 # In other series, the leads to False, so do that here too
                 ret[na_mask] = False

pandas/tests/test_categorical.py

@@ -3821,6 +3821,43 @@ def test_cat_equality(self):
         pytest.raises(TypeError, lambda: a > b)
         pytest.raises(TypeError, lambda: b > a)
 
+    @pytest.mark.parametrize('ctor', [
+        lambda *args, **kwargs: Categorical(*args, **kwargs),
+        lambda *args, **kwargs: Series(Categorical(*args, **kwargs)),
+    ])
+    def test_unordered_different_order_equal(self, ctor):
+        # https://github.com/pandas-dev/pandas/issues/16014
+        c1 = ctor(['a', 'b'], categories=['a', 'b'], ordered=False)
+        c2 = ctor(['a', 'b'], categories=['b', 'a'], ordered=False)
+        assert (c1 == c2).all()
+
+        c1 = ctor(['a', 'b'], categories=['a', 'b'], ordered=False)
+        c2 = ctor(['b', 'a'], categories=['b', 'a'], ordered=False)
+        assert (c1 != c2).all()
+
+        c1 = ctor(['a', 'a'], categories=['a', 'b'], ordered=False)
+        c2 = ctor(['b', 'b'], categories=['b', 'a'], ordered=False)
+        assert (c1 != c2).all()
+
+        c1 = ctor(['a', 'a'], categories=['a', 'b'], ordered=False)
+        c2 = ctor(['a', 'b'], categories=['b', 'a'], ordered=False)
+        result = c1 == c2
+        tm.assert_numpy_array_equal(np.array(result), np.array([True, False]))
+
+    def test_unordered_different_categories_raises(self):
+        c1 = Categorical(['a', 'b'], categories=['a', 'b'], ordered=False)
+        c2 = Categorical(['a', 'c'], categories=['c', 'a'], ordered=False)
+        with tm.assert_raises_regex(TypeError,
+                                    "Categoricals can only be compared"):
+            c1 == c2
+
+    def test_compare_different_lengths(self):
+        c1 = Categorical([], categories=['a', 'b'])
+        c2 = Categorical([], categories=['a'])
+        msg = "Categories are different lengths"
+        with tm.assert_raises_regex(TypeError, msg):
+            c1 == c2
+
     def test_concat_append(self):
         cat = pd.Categorical(["a", "b"], categories=["a", "b"])
         vals = [1, 2]