BUG: `DateTimeIndex + DateTimeIndex.freq` giving incorrect result if timezone is set.
- [x] I have checked that this issue has not already been reported.

- [x] 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.

---
#### Code Sample, a copy-pastable example

```python
import pandas as pd
i = pd.date_range('2020-03-27', freq='D', periods=4, tz='Europe/Berlin')
i2 = i + i.freq
i2.freq = 'D' # ValueError: Inferred frequency None from passed values does not conform to passed frequency D
```

#### Problem description

`i` is as expected.
```
DatetimeIndex(['2020-03-27 00:00:00+01:00', '2020-03-28 00:00:00+01:00',
               '2020-03-29 00:00:00+01:00', '2020-03-30 00:00:00+02:00'],
              dtype='datetime64[ns, Europe/Berlin]', freq='D') #good
```
but `i2` has two issues: 
```
DatetimeIndex(['2020-03-29 00:00:00+01:00', '2020-03-30 01:00:00+02:00',
               '2020-03-31 00:00:00+02:00', '2020-04-01 00:00:00+02:00'],
              dtype='datetime64[ns, Europe/Berlin]', freq='D') 
#issue: 2020-03-30 timestamp not at midnight
#issue: i2 has attribute `freq='D'`, but pd.infer_freq returns None
```

#### Expected Output

`i2` to be 
```
DatetimeIndex(['2020-03-29 00:00:00+01:00', '2020-03-30 00:00:00+02:00',
               '2020-03-31 00:00:00+02:00', '2020-04-01 00:00:00+02:00'],
              dtype='datetime64[ns, Europe/Berlin]', freq='D') 
```
(`2020-03-30 00:00:00+02:00` is only change),
with `i2.freq` not returning `None`

#### Related
May be related to [#38243](https://github.com/pandas-dev/pandas/issues/38243)

#### Output of ``pd.show_versions()``

<details>

INSTALLED VERSIONS
------------------
commit           : 3e89b4c4b1580aa890023fc550774e63d499da25
python           : 3.8.5.final.0
python-bits      : 64
OS               : Windows
OS-release       : 10
Version          : 10.0.18362
machine          : AMD64
processor        : Intel64 Family 6 Model 158 Stepping 10, GenuineIntel
byteorder        : little
LC_ALL           : None
LANG             : en
LOCALE           : de_DE.cp1252

pandas           : 1.2.0
numpy            : 1.19.5
pytz             : 2020.5
dateutil         : 2.8.1
pip              : 20.3.3
setuptools       : 51.1.2.post20210112
Cython           : None
pytest           : 6.2.1
hypothesis       : None
sphinx           : 3.4.3
blosc            : None
feather          : None
xlsxwriter       : None
lxml.etree       : None
html5lib         : None
pymysql          : None
psycopg2         : None
jinja2           : 2.11.2
IPython          : 7.12.0
pandas_datareader: None
bs4              : None
bottleneck       : None
fsspec           : None
fastparquet      : None
gcsfs            : None
matplotlib       : 3.3.3
numexpr          : None
odfpy            : None
openpyxl         : None
pandas_gbq       : None
pyarrow          : None
pyxlsb           : None
s3fs             : None
scipy            : 1.6.0
sqlalchemy       : 1.3.22
tables           : None
tabulate         : None
xarray           : None
xlrd             : None
xlwt             : None
numba            : None
</details>
