i1 = pd.date_range('2020-03-27', periods=5, freq='D', tz='Europe/Berlin')
i2 = pd.date_range('2020-03-30', periods=5, freq='D', tz='Europe/Berlin')
pd.Index([i for i in i1 if i in i2], freq=i1.freq)

>>> i1 = pd.date_range('2020-03-27', periods=5, freq='D', tz='Europe/Berlin')
>>> i2 = pd.date_range('2020-03-30', periods=5, freq='D', tz='Europe/Berlin')
>>> i1.intersection(i2)
DatetimeIndex([], dtype='datetime64[ns, Europe/Berlin]', freq='D')

>>> i1 = pd.date_range('2020-03-20', periods=5, freq='D', tz='Europe/Berlin')
>>> i2 = pd.date_range('2020-03-23', periods=5, freq='D', tz='Europe/Berlin')
>>> i1.intersection(i2)
DatetimeIndex(['2020-03-23 00:00:00+01:00', '2020-03-24 00:00:00+01:00'], dtype='datetime64[ns, Europe/Berlin]', freq='D')

>>> pd.show_versions()

INSTALLED VERSIONS
------------------
commit           : 83f0d71a2cbde738c459e4feca54288b77d00e3d
python           : 3.9.12.final.0
python-bits      : 64
OS               : Windows
OS-release       : 10
Version          : 10.0.19044
machine          : AMD64
processor        : Intel64 Family 6 Model 165 Stepping 2, GenuineIntel
byteorder        : little
LC_ALL           : None
LANG             : None
LOCALE           : de_DE.cp1252

pandas           : 1.5.0.dev0+666.g83f0d71a2c
numpy            : 1.22.3
pytz             : 2022.1
dateutil         : 2.8.2
pip              : 21.2.4
setuptools       : 58.0.4
Cython           : None
pytest           : None
hypothesis       : None
sphinx           : None
blosc            : None
feather          : None
xlsxwriter       : None
lxml.etree       : None
html5lib         : None
pymysql          : None
psycopg2         : None
jinja2           : None
IPython          : None
pandas_datareader: None
bs4              : None
bottleneck       : None
brotli           : None
fastparquet      : None
fsspec           : None
gcsfs            : None
markupsafe       : None
matplotlib       : None
numba            : None
numexpr          : None
odfpy            : None
openpyxl         : None
pandas_gbq       : None
pyarrow          : None
pyreadstat       : None
pyxlsb           : None
s3fs             : None
scipy            : None
snappy           : None
sqlalchemy       : None
tables           : None
tabulate         : None
xarray           : None
xlrd             : None
xlwt             : None
zstandard        : None

def intersection(*indices: pd.DatetimeIndex) -> pd.DatetimeIndex:
    """Intersect several DatetimeIndices.

    Parameters
    ----------
    *indices : pd.DatetimeIndex
        The indices to intersect.

    Returns
    -------
    pd.DatetimeIndex
        The intersection, i.e., datetimeindex with values that exist in each index.

    Notes
    -----
    If indices have distinct timezones or names, the values from the first index are used.
    """
    if len(indices) == 0:
        raise ValueError("Must specify at least one index.")

    distinct_freqs = set([i.freq for i in indices])
    if len(indices) > 1 and len(distinct_freqs) != 1:
        raise ValueError(
            f"Indices must not have equal frequencies; got {distinct_freqs}."
        )

    tznaive = [i.tz is None for i in indices]
    if any(tznaive) and not all(tznaive):
        raise ValueError(
            f"All indices must be either timezone-aware or timezone-naive; got {tznaive} naive from {len(indices)}."
        )

    freq, name, tz = indices[0].freq, indices[0].name, indices[0].tz

    # Calculation is cumbersome: pandas DatetimeIndex.intersection not working correctly on timezone-aware indices.
    values = set(indices[0])
    for idx in indices[1:]:
        values = values.intersection(set(idx))
    idx = pd.DatetimeIndex(sorted(list(values)), freq=freq, name=name, tz=tz)

    return idx