diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
index 09fe885e47754..ad5de61cf665b 100644
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -2828,8 +2828,8 @@ def intersection(self, other, sort=False):
 
         if not is_dtype_equal(self.dtype, other.dtype):
             dtype = find_common_type([self.dtype, other.dtype])
-            this = self.astype(dtype)
-            other = other.astype(dtype)
+            this = self.astype(dtype, copy=False)
+            other = other.astype(dtype, copy=False)
             return this.intersection(other, sort=sort)
 
         result = self._intersection(other, sort=sort)
diff --git a/pandas/core/indexes/interval.py b/pandas/core/indexes/interval.py
index bd92926941aa1..8af9d12e22ee0 100644
--- a/pandas/core/indexes/interval.py
+++ b/pandas/core/indexes/interval.py
@@ -967,9 +967,20 @@ def intersection(self, other, sort=False) -> Index:
         self._assert_can_do_setop(other)
         other, _ = self._convert_can_do_setop(other)
 
+        if self.equals(other) and not self.has_duplicates:
+            return self._get_reconciled_name_object(other)
+
         if not isinstance(other, IntervalIndex):
             return self.astype(object).intersection(other)
 
+        result = self._intersection(other, sort=sort)
+        return self._wrap_setop_result(other, result)
+
+    def _intersection(self, other, sort):
+        """
+        intersection specialized to the case with matching dtypes.
+        """
+        # For IntervalIndex we also know other.closed == self.closed
         if self.left.is_unique and self.right.is_unique:
             taken = self._intersection_unique(other)
         elif other.left.is_unique and other.right.is_unique and self.isna().sum() <= 1:
@@ -983,7 +994,7 @@ def intersection(self, other, sort=False) -> Index:
         if sort is None:
             taken = taken.sort_values()
 
-        return self._wrap_setop_result(other, taken)
+        return taken
 
     def _intersection_unique(self, other: "IntervalIndex") -> "IntervalIndex":
         """
diff --git a/pandas/core/indexes/multi.py b/pandas/core/indexes/multi.py
index 4aedf03ca1800..03f6387ae2dc5 100644
--- a/pandas/core/indexes/multi.py
+++ b/pandas/core/indexes/multi.py
@@ -3723,7 +3723,7 @@ def _convert_can_do_setop(self, other):
                 try:
                     other = MultiIndex.from_tuples(other)
                 except (ValueError, TypeError) as err:
-                    # ValueError raised by tupels_to_object_array if we
+                    # ValueError raised by tuples_to_object_array if we
                     #  have non-object dtype
                     raise TypeError(msg) from err
         else: