Digital Clock Times Puzzle - Solution

The Puzzle:

Tim saw that his oven digital clock was showing 1:23, and noticed that each number was one higher than the number to its left.

He wondered: of all the different clock times from 0:00 to 12:59, how many are like that? And what exactly are those clock times?

Can you help Tim out?

We may have missed one or two, but this is what we got:

0:12
1:23
2:34
3:45
4:56
12:34

Can you do better?

Note 1: Some clocks display all 4 digits, so would only have 01:23 and 12:34
Note 2: a 24-hour clock could also have 23:45

Puzzle Author: Mitchell Marcus