handle NaT add/sub in one place
State: closed

ATM DatetimeIndex/TimedeltaIndex/PeriodIndex each have special-casing for addition/subtraction of NaT.  This checks for NaT up-front and puts all of that logic into add_nat and sub_nat methods.

Also makes another step towards centralizing all of the wrapping of the add/sub ops at the end of `__add__` and `__sub__`.

Should be orthogonal to other outstanding PRs, albeit just barely.

Labels:
name: Datetime
description: Datetime data dtype
name: Missing-data
description: np.nan, pd.NaT, pd.NA, dropna, isnull, interpolate

Comments:
# [Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/19903?src=pr&el=h1) Report
> Merging [#19903](https://codecov.io/gh/pandas-dev/pandas/pull/19903?src=pr&el=desc) into [master](https://codecov.io/gh/pandas-dev/pandas/commit/1e4c50a56f7e953ab84308f000dff6fc1ac71171?src=pr&el=desc) will **increase** coverage by `<.01%`.
> The diff coverage is `87.5%`.

[![Impacted file tree graph](https://codecov.io/gh/pandas-dev/pandas/pull/19903/graphs/tree.svg?width=650&token=eZ4WkYLtcO&height=150&src=pr)](https://codecov.io/gh/pandas-dev/pandas/pull/19903?src=pr&el=tree)

```diff
@@            Coverage Diff             @@
##           master   #19903      +/-   ##
==========================================
+ Coverage   91.66%   91.67%   +<.01%     
==========================================
  Files         150      150              
  Lines       48969    48967       -2     
==========================================
+ Hits        44886    44889       +3     
+ Misses       4083     4078       -5
```

| Flag | Coverage Δ | |
|---|---|---|
| #multiple | `90.05% <87.5%> (ø)` | :arrow_up: |
| #single | `41.85% <31.25%> (ø)` | :arrow_up: |

| [Impacted Files](https://codecov.io/gh/pandas-dev/pandas/pull/19903?src=pr&el=tree) | Coverage Δ | |
|---|---|---|
| [pandas/core/indexes/timedeltas.py](https://codecov.io/gh/pandas-dev/pandas/pull/19903/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvaW5kZXhlcy90aW1lZGVsdGFzLnB5) | `91.17% <100%> (-0.04%)` | :arrow_down: |
| [pandas/core/indexes/period.py](https://codecov.io/gh/pandas-dev/pandas/pull/19903/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvaW5kZXhlcy9wZXJpb2QucHk=) | `94.65% <100%> (+1.3%)` | :arrow_up: |
| [pandas/core/indexes/datetimes.py](https://codecov.io/gh/pandas-dev/pandas/pull/19903/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvaW5kZXhlcy9kYXRldGltZXMucHk=) | `95.5% <75%> (-0.11%)` | :arrow_down: |
| [pandas/core/indexes/datetimelike.py](https://codecov.io/gh/pandas-dev/pandas/pull/19903/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvaW5kZXhlcy9kYXRldGltZWxpa2UucHk=) | `96.83% <86.95%> (-0.5%)` | :arrow_down: |

------

[Continue to review full report at Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/19903?src=pr&el=continue).
> **Legend** - [Click here to learn more](https://docs.codecov.io/docs/codecov-delta)
> `Δ = absolute <relative> (impact)`, `ø = not affected`, `? = missing data`
> Powered by [Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/19903?src=pr&el=footer). Last update [1e4c50a...244bdaa](https://codecov.io/gh/pandas-dev/pandas/pull/19903?src=pr&el=lastupdated). Read the [comment docs](https://docs.codecov.io/docs/pull-request-comments).

thanks