BUG: RangeIndex.union (#44019)

Author: jbrockmendel

doc/source/whatsnew/v1.4.0.rst

@@ -578,6 +578,7 @@ Styler
 Other
 ^^^^^
 - Bug in :meth:`CustomBusinessMonthBegin.__add__` (:meth:`CustomBusinessMonthEnd.__add__`) not applying the extra ``offset`` parameter when beginning (end) of the target month is already a business day (:issue:`41356`)
+- Bug in :meth:`RangeIndex.union` with another ``RangeIndex`` with matching (even) ``step`` and starts differing by strictly less than ``step / 2`` (:issue:`44019`)
 
 .. ***DO NOT USE THIS SECTION***
 

pandas/core/indexes/range.py

@@ -635,10 +635,13 @@ def _union(self, other: Index, sort):
                     return type(self)(start_r, end_r + step_s, step_s)
                 if (
                     (step_s % 2 == 0)
-                    and (abs(start_s - start_o) <= step_s / 2)
-                    and (abs(end_s - end_o) <= step_s / 2)
+                    and (abs(start_s - start_o) == step_s / 2)
+                    and (abs(end_s - end_o) == step_s / 2)
                 ):
+                    # e.g. range(0, 10, 2) and range(1, 11, 2)
+                    #  but not range(0, 20, 4) and range(1, 21, 4) GH#44019
                     return type(self)(start_r, end_r + step_s / 2, step_s / 2)
+
             elif step_o % step_s == 0:
                 if (
                     (start_o - start_s) % step_s == 0

pandas/tests/indexes/ranges/test_setops.py

@@ -290,6 +290,15 @@ def test_union_sorted(self, unions):
         tm.assert_index_equal(res2, expected_sorted, exact=True)
         tm.assert_index_equal(res3, expected_sorted, exact="equiv")
 
+    def test_union_same_step_misaligned(self):
+        # GH#44019
+        left = RangeIndex(range(0, 20, 4))
+        right = RangeIndex(range(1, 21, 4))
+
+        result = left.union(right)
+        expected = Int64Index([0, 1, 4, 5, 8, 9, 12, 13, 16, 17])
+        tm.assert_index_equal(result, expected, exact=True)
+
     def test_difference(self):
         # GH#12034 Cases where we operate against another RangeIndex and may
         #  get back another RangeIndex