BUG: PeriodDtype incorrectly caching different DateOffset freqs

doc/source/whatsnew/v2.1.0.rst

@@ -376,12 +376,12 @@ Period
 ^^^^^^
 - :meth:`PeriodIndex.map` with ``na_action="ignore"`` now works as expected (:issue:`51644`)
 - Bug in :class:`PeriodDtype` constructor failing to raise ``TypeError`` when no argument is passed or when ``None`` is passed (:issue:`27388`)
+- Bug in :class:`PeriodDtype` constructor incorrectly returning the same ``normalize`` for different :class:`DateOffset` ``freq`` inputs (:issue:`24121`)
 - Bug in :class:`PeriodDtype` constructor raising ``ValueError`` instead of ``TypeError`` when an invalid type is passed (:issue:`51790`)
 - Bug in :func:`read_csv` not processing empty strings as a null value, with ``engine="pyarrow"`` (:issue:`52087`)
 - Bug in :func:`read_csv` returning ``object`` dtype columns instead of ``float64`` dtype columns with ``engine="pyarrow"`` for columns that are all null with ``engine="pyarrow"`` (:issue:`52087`)
 - Bug in :meth:`arrays.PeriodArray.map` and :meth:`PeriodIndex.map`, where the supplied callable operated array-wise instead of element-wise (:issue:`51977`)
 - Bug in incorrectly allowing construction of :class:`Period` or :class:`PeriodDtype` with :class:`CustomBusinessDay` freq; use :class:`BusinessDay` instead (:issue:`52534`)
--
 
 Plotting
 ^^^^^^^^

pandas/core/dtypes/dtypes.py

@@ -899,7 +899,10 @@ class PeriodDtype(PeriodDtypeBase, PandasExtensionDtype):
     num = 102
     _metadata = ("freq",)
     _match = re.compile(r"(P|p)eriod\[(?P<freq>.+)\]")
-    _cache_dtypes: dict[str_type, PandasExtensionDtype] = {}
+    # error: Incompatible types in assignment (expression has type
+    # "Dict[int, PandasExtensionDtype]", base class "PandasExtensionDtype"
+    # defined the type as "Dict[str, PandasExtensionDtype]")  [assignment]
+    _cache_dtypes: dict[BaseOffset, PeriodDtype] = {}  # type: ignore[assignment] # noqa:E501
     __hash__ = PeriodDtypeBase.__hash__
     _freq: BaseOffset
 
@@ -916,12 +919,12 @@ def __new__(cls, freq):
             freq = cls._parse_dtype_strict(freq)
 
         try:
-            return cls._cache_dtypes[freq.freqstr]
+            return cls._cache_dtypes[freq]
         except KeyError:
             dtype_code = freq._period_dtype_code
             u = PeriodDtypeBase.__new__(cls, dtype_code, freq.n)
             u._freq = freq
-            cls._cache_dtypes[freq.freqstr] = u
+            cls._cache_dtypes[freq] = u
             return u
 
     def __reduce__(self):

pandas/tests/dtypes/test_dtypes.py

@@ -559,6 +559,14 @@ def test_not_string(self):
         # though PeriodDtype has object kind, it cannot be string
         assert not is_string_dtype(PeriodDtype("D"))
 
+    def test_perioddtype_caching_dateoffset_normalize(self):
+        # GH 24121
+        per_d = PeriodDtype(pd.offsets.YearEnd(normalize=True))
+        assert per_d.freq.normalize
+
+        per_d2 = PeriodDtype(pd.offsets.YearEnd(normalize=False))
+        assert not per_d2.freq.normalize
+
 
 class TestIntervalDtype(Base):
     @pytest.fixture