API: Change Period('NAT') to return NaT

closes #12759
closes #13582

Author: sinhrks <[email protected]>

Closes #13609 from sinhrks/period_nat and squashes the following commits:

9305c36 [sinhrks] COMPAT: Period(NaT) now returns pd.NaT

Author: sinhrks

doc/source/whatsnew/v0.19.0.txt

@@ -446,6 +446,45 @@ Furthermore:
 - Passing duplicated ``percentiles`` will now raise a ``ValueError``.
 - Bug in ``.describe()`` on a DataFrame with a mixed-dtype column index, which would previously raise a ``TypeError`` (:issue:`13288`)
 
+.. _whatsnew_0190.api.periodnat:
+
+``Period('NaT')`` now returns ``pd.NaT``
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Previously, ``Period`` has its own ``Period('NaT')`` representation different from ``pd.NaT``. Now ``Period('NaT')`` has been changed to return ``pd.NaT``. (:issue:`12759`, :issue:`13582`)
+
+Previous Behavior:
+
+.. code-block:: ipython
+
+   In [5]: pd.Period('NaT', freq='D')
+   Out[5]: Period('NaT', 'D')
+
+New Behavior:
+
+.. ipython:: python
+
+   pd.Period('NaT')
+
+
+To be compat with ``Period`` addition and subtraction, ``pd.NaT`` now supports addition and subtraction with ``int``. Previously it raises ``ValueError``.
+
+Previous Behavior:
+
+.. code-block:: ipython
+
+   In [5]: pd.NaT + 1
+   ...
+   ValueError: Cannot add integral value to Timestamp without freq.
+
+New Behavior:
+
+.. ipython:: python
+
+   pd.NaT + 1
+   pd.NaT - 1
+
+
 .. _whatsnew_0190.deprecations:
 
 Deprecations