BUG: IntervalIndex is_non_overlapping_monotonic is incorrect when endpoints are closed on both sides
State: closed

 - [x] closes #16560
 - [x] tests added / passed
 - [x] passes ``git diff upstream/master --name-only -- '*.py' | flake8 --diff``
 - [x] whatsnew entry


Labels:
name: Bug
name: Interval
description: Interval data type

Comments:
# [Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/16588?src=pr&el=h1) Report
> Merging [#16588](https://codecov.io/gh/pandas-dev/pandas/pull/16588?src=pr&el=desc) into [master](https://codecov.io/gh/pandas-dev/pandas/commit/31e67d5d834bd1754e1ce5cdf6725c54bf06a01d?src=pr&el=desc) will **increase** coverage by `<.01%`.
> The diff coverage is `100%`.

[![Impacted file tree graph](https://codecov.io/gh/pandas-dev/pandas/pull/16588/graphs/tree.svg?height=150&token=eZ4WkYLtcO&src=pr&width=650)](https://codecov.io/gh/pandas-dev/pandas/pull/16588?src=pr&el=tree)

```diff
@@            Coverage Diff             @@
##           master   #16588      +/-   ##
==========================================
+ Coverage   90.75%   90.75%   +<.01%     
==========================================
  Files         161      161              
  Lines       51097    51099       +2     
==========================================
+ Hits        46372    46374       +2     
  Misses       4725     4725
```

| Flag | Coverage Δ | |
|---|---|---|
| #multiple | `88.59% <100%> (ø)` | :arrow_up: |
| #single | `40.16% <0%> (-0.01%)` | :arrow_down: |

| [Impacted Files](https://codecov.io/gh/pandas-dev/pandas/pull/16588?src=pr&el=tree) | Coverage Δ | |
|---|---|---|
| [pandas/core/indexes/interval.py](https://codecov.io/gh/pandas-dev/pandas/pull/16588?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvaW5kZXhlcy9pbnRlcnZhbC5weQ==) | `92.61% <100%> (+0.02%)` | :arrow_up: |

------

[Continue to review full report at Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/16588?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/16588?src=pr&el=footer). Last update [31e67d5...bdf1146](https://codecov.io/gh/pandas-dev/pandas/pull/16588?src=pr&el=lastupdated). Read the [comment docs](https://docs.codecov.io/docs/pull-request-comments).

# [Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/16588?src=pr&el=h1) Report
> Merging [#16588](https://codecov.io/gh/pandas-dev/pandas/pull/16588?src=pr&el=desc) into [master](https://codecov.io/gh/pandas-dev/pandas/commit/31e67d5d834bd1754e1ce5cdf6725c54bf06a01d?src=pr&el=desc) will **increase** coverage by `<.01%`.
> The diff coverage is `100%`.

[![Impacted file tree graph](https://codecov.io/gh/pandas-dev/pandas/pull/16588/graphs/tree.svg?src=pr&token=eZ4WkYLtcO&height=150&width=650)](https://codecov.io/gh/pandas-dev/pandas/pull/16588?src=pr&el=tree)

```diff
@@            Coverage Diff             @@
##           master   #16588      +/-   ##
==========================================
+ Coverage   90.75%   90.75%   +<.01%     
==========================================
  Files         161      161              
  Lines       51097    51099       +2     
==========================================
+ Hits        46372    46374       +2     
  Misses       4725     4725
```

| Flag | Coverage Δ | |
|---|---|---|
| #multiple | `88.59% <100%> (ø)` | :arrow_up: |
| #single | `40.16% <0%> (-0.01%)` | :arrow_down: |

| [Impacted Files](https://codecov.io/gh/pandas-dev/pandas/pull/16588?src=pr&el=tree) | Coverage Δ | |
|---|---|---|
| [pandas/core/indexes/interval.py](https://codecov.io/gh/pandas-dev/pandas/pull/16588?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvaW5kZXhlcy9pbnRlcnZhbC5weQ==) | `92.61% <100%> (+0.02%)` | :arrow_up: |

------

[Continue to review full report at Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/16588?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/16588?src=pr&el=footer). Last update [31e67d5...bdf1146](https://codecov.io/gh/pandas-dev/pandas/pull/16588?src=pr&el=lastupdated). Read the [comment docs](https://docs.codecov.io/docs/pull-request-comments).

Fixed the issues from the review, except for changing ```==``` to ```is``` in the asserts. Using ```is``` resulted in assert errors so I kept it the way it was originally.

Added additional tests and updated the whatsnew section in the docs. Thanks.
ping @jreback : This can be closed, as the issue was resolved by #17238
@jschendel : Confirmed and closing.