BUG: DatetimeIndex.intersection gives incorrect result

[grotmol]

• I have checked that this issue has not already been reported.

• I have confirmed this bug exists on the latest version of pandas.

• (optional) I have confirmed this bug exists on the master branch of pandas.

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Note: Please read this guide detailing how to provide the necessary information for us to reproduce your bug.

Code Sample, a copy-pastable example

import pandas as pd
index = pd.DatetimeIndex(['2018-12-31', '2019-03-31', '2019-06-30', '2019-09-30', '2019-12-31', '2020-03-31'], freq='Q-DEC')
print(index[::2].intersection(index[1::2]))

Problem description

The produced output is:
DatetimeIndex(['2019-06-30', '2019-12-31'], dtype='datetime64[ns]', freq='2Q-DEC')

This is clearly incorrect, as there is no overlap between the indices index[::2] and index[1::2].

The issue is related to the freq attribute, which is set to '2Q-DEC' for both index[::2] and index[1::2]. If I remove or change the freq attribute, then it works correctly.

Expected Output

DatetimeIndex([], dtype='datetime64[ns]', freq=None)

Output of pd.show_versions()

INSTALLED VERSIONS

commit : 2cb9652
python : 3.8.5.final.0
python-bits : 64
OS : Darwin
OS-release : 20.5.0
Version : Darwin Kernel Version 20.5.0: Sat May 8 05:10:33 PDT 2021; root:xnu-7195.121.3~9/RELEASE_X86_64
machine : x86_64
processor : i386
byteorder : little
LC_ALL : en_US.UTF-8
LANG : en_US.UTF-8
LOCALE : en_US.UTF-8

pandas : 1.2.4
numpy : 1.20.3
pytz : 2021.1
dateutil : 2.8.1
pip : 21.0.1
setuptools : 52.0.0
Cython : 0.29.17
pytest : None
hypothesis : None
sphinx : None
blosc : None
feather : 0.4.1
xlsxwriter : 1.4.3
lxml.etree : 4.6.3
html5lib : None
pymysql : None
psycopg2 : 2.8.6 (dt dec pq3 ext lo64)
jinja2 : None
IPython : 7.21.0
pandas_datareader: 0.9.0
bs4 : None
bottleneck : None
fsspec : None
fastparquet : None
gcsfs : None
matplotlib : 3.3.4
numexpr : None
odfpy : None
openpyxl : 3.0.6
pandas_gbq : None
pyarrow : 3.0.0
pyxlsb : None
s3fs : None
scipy : 1.4.1
sqlalchemy : 1.3.23
tables : None
tabulate : 0.8.9
xarray : None
xlrd : None
xlwt : 1.3.0
numba : 0.52.0

[simonjayhawkins]

Expected Output

DatetimeIndex([], dtype='datetime64[ns]', freq=None)

This was the output in pandas 1.0.5

first bad commit: [29c820f] BUG: DTI/TDI intersection result names (#33904)

cc @jbrockmendel

[jbrockmendel]

changing _can_fast_intersect to return self.freq.n == 1 on the last line appears to solve this. still need to run the full test suite

[grotmol]

@simonjayhawkins @jbrockmendel: Thanks for fixing this so swiftly!