[math-fun] An insomniac's problem

There's a digital clock at my bedside that shows hours minutes and seconds, and which I usually look at the moment I wake up. A few days ago, I happened to wake up at a palindromic time, and out of interest, waited until the next palindromic time, and noticed the interval between them. Last night exactly the same thing happened, but the interval was exactly four times as long as in the previous case. What did the clock read when I woke up last night? John Conway