Now, there are two cases:
1) The minute hand and hour hand are the mirror images of themselves.
2) The minute hand and hour hand are the mirror images of each other, but not of themselves. (The but prevents noon from being counted twice)
Case 1 happens iff both hands are at either 12 or 6. The hour hand is only at 12 once and at 6 once, so it can happen at most twice. Fortunately, both these times the minute hand will be at 12, so we have two instances of case 1.
Let a rough hour be the time defined only from the hour number on a digital clock. For example, the rough hour 4 is the time starting at 4:00 and ending just before 5:00. Case 2 happens exactly once every rough hour. To see why, consider the mirror image of the hour hand as it moves counter-clockwise slowly and continuously. The minute hand makes a continuous clockwise sweep around the clock, and so it must meet the mirrored hour hand. The mirrored hour hand has nowhere to run and nowhere to hide.
So we have 2 instances of case 1, and 12 instances of case 2. That makes a total of 14.