diff --git a/pandas/core/arrays/interval.py b/pandas/core/arrays/interval.py
index 3263dd73fe4dc..9c2e85c4df564 100644
--- a/pandas/core/arrays/interval.py
+++ b/pandas/core/arrays/interval.py
@@ -764,7 +764,7 @@ def _cmp_method(self, other, op):
                 if self.closed != other.categories.closed:
                     return invalid_comparison(self, other, op)
 
-                other = other.categories.take(
+                other = other.categories._values.take(
                     other.codes, allow_fill=True, fill_value=other.categories._na_value
                 )
 
diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
index 796aadf9e4061..2305c1cbb698e 100644
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -2172,11 +2172,6 @@ def _drop_level_numbers(self, levnums: list[int]):
     @final
     def _can_hold_na(self) -> bool:
         if isinstance(self.dtype, ExtensionDtype):
-            if isinstance(self.dtype, IntervalDtype):
-                # FIXME(GH#45720): this is inaccurate for integer-backed
-                #  IntervalArray, but without it other.categories.take raises
-                #  in IntervalArray._cmp_method
-                return True
             return self.dtype._can_hold_na
         if self.dtype.kind in "iub":
             return False