At the time shown, the mirror image of the clock reads exactly 2 hours LATER than the real time. At which other time on the dial does this happen?

The clock shows 5:00 and its image 7:00. The image of a time $T$ (minutes past 12) is $720-T$; we need $720-T=T+120$ on a 12-hour dial, i.e. $2T\equiv 600(\mathrm{mod}720)$, so $T=300$ (5:00) or 660 (11:00). At 11:00 the image shows 1:00, two hours later. (At 7:00 the image is 5:00, two hours EARLIER.)