diff --git a/pandas/_libs/interval.pyx b/pandas/_libs/interval.pyx
index f8bcbcfb158b5..10becdce5d6dd 100644
--- a/pandas/_libs/interval.pyx
+++ b/pandas/_libs/interval.pyx
@@ -179,7 +179,8 @@ cdef class IntervalMixin:
         return (self.right == self.left) & (self.closed != 'both')
 
     def _check_closed_matches(self, other, name='other'):
-        """Check if the closed attribute of `other` matches.
+        """
+        Check if the closed attribute of `other` matches.
 
         Note that 'left' and 'right' are considered different from 'both'.
 
diff --git a/pandas/core/indexes/interval.py b/pandas/core/indexes/interval.py
index 267d1330faceb..21264b00b91f8 100644
--- a/pandas/core/indexes/interval.py
+++ b/pandas/core/indexes/interval.py
@@ -130,19 +130,13 @@ def wrapped(self, other, sort=False):
             if op_name in ("difference",):
                 result = result.astype(self.dtype)
             return result
-        elif self.closed != other.closed:
-            raise ValueError(
-                "can only do set operations between two IntervalIndex "
-                "objects that are closed on the same side"
-            )
 
-        # GH 19016: ensure set op will not return a prohibited dtype
-        subtypes = [self.dtype.subtype, other.dtype.subtype]
-        common_subtype = find_common_type(subtypes)
-        if is_object_dtype(common_subtype):
+        if self._is_non_comparable_own_type(other):
+            # GH#19016: ensure set op will not return a prohibited dtype
             raise TypeError(
-                f"can only do {op_name} between two IntervalIndex "
-                "objects that have compatible dtypes"
+                "can only do set operations between two IntervalIndex "
+                "objects that are closed on the same side "
+                "and have compatible dtypes"
             )
 
         return method(self, other, sort)
@@ -717,11 +711,8 @@ def get_indexer(
             if self.equals(target_as_index):
                 return np.arange(len(self), dtype="intp")
 
-            # different closed or incompatible subtype -> no matches
-            common_subtype = find_common_type(
-                [self.dtype.subtype, target_as_index.dtype.subtype]
-            )
-            if self.closed != target_as_index.closed or is_object_dtype(common_subtype):
+            if self._is_non_comparable_own_type(target_as_index):
+                # different closed or incompatible subtype -> no matches
                 return np.repeat(np.intp(-1), len(target_as_index))
 
             # non-overlapping -> at most one match per interval in target_as_index
@@ -763,10 +754,8 @@ def get_indexer_non_unique(
 
         # check that target_as_index IntervalIndex is compatible
         if isinstance(target_as_index, IntervalIndex):
-            common_subtype = find_common_type(
-                [self.dtype.subtype, target_as_index.dtype.subtype]
-            )
-            if self.closed != target_as_index.closed or is_object_dtype(common_subtype):
+
+            if self._is_non_comparable_own_type(target_as_index):
                 # different closed or incompatible subtype -> no matches
                 return (
                     np.repeat(-1, len(target_as_index)),
@@ -837,6 +826,16 @@ def _convert_list_indexer(self, keyarr):
 
         return locs
 
+    def _is_non_comparable_own_type(self, other: "IntervalIndex") -> bool:
+        # different closed or incompatible subtype -> no matches
+
+        # TODO: once closed is part of IntervalDtype, we can just define
+        #  is_comparable_dtype GH#19371
+        if self.closed != other.closed:
+            return True
+        common_subtype = find_common_type([self.dtype.subtype, other.dtype.subtype])
+        return is_object_dtype(common_subtype)
+
     # --------------------------------------------------------------------
 
     @cache_readonly
diff --git a/pandas/tests/indexes/interval/test_setops.py b/pandas/tests/indexes/interval/test_setops.py
index 562497b29af12..0b94d70367b4d 100644
--- a/pandas/tests/indexes/interval/test_setops.py
+++ b/pandas/tests/indexes/interval/test_setops.py
@@ -159,18 +159,18 @@ def test_set_incompatible_types(self, closed, op_name, sort):
         # mixed closed
         msg = (
             "can only do set operations between two IntervalIndex objects "
-            "that are closed on the same side"
+            "that are closed on the same side and have compatible dtypes"
         )
         for other_closed in {"right", "left", "both", "neither"} - {closed}:
             other = monotonic_index(0, 11, closed=other_closed)
-            with pytest.raises(ValueError, match=msg):
+            with pytest.raises(TypeError, match=msg):
                 set_op(other, sort=sort)
 
         # GH 19016: incompatible dtypes
         other = interval_range(Timestamp("20180101"), periods=9, closed=closed)
         msg = (
-            f"can only do {op_name} between two IntervalIndex objects that have "
-            "compatible dtypes"
+            "can only do set operations between two IntervalIndex objects "
+            "that are closed on the same side and have compatible dtypes"
         )
         with pytest.raises(TypeError, match=msg):
             set_op(other, sort=sort)