The Ambiguous Watch

HELLO PEOPLE :D

Here is a lovely little problem for you to ponder over, if you've managed to finish all your holiday homework! I thought of it a short while back.

Last year, Yongquan went to Macau on holiday. Over there, he bought a watch, because it was retro and pretty. Unfortunately, because the watch is supposed to be retro and pretty, the watchmakers did away with the second hand, and the hour and minute hands are identical and indistinguishable. Many a time did Yongquan despair because he was confused when reading the time.

"Aha!" said Yongquan's mother, when he told her of his dilemma. "Notice that at say, three o'clock, one hand will be pointing straight at 12 and the other at 3. It then can only be three o'clock, because the alternative is meaningless. Using elimination of this variety, you should be able to tell the time with no problem at all!"

"But!" said Yongquan's father, who is sharp like a needle. "There are several moments in a day when the alternative configuration depicts a valid time. When that happens, you should ... just check the time on your mobile phone."

How many times in a day will both configurations of the hour and minute hand of the watch depict valid times, and what are they?