Not sure what you're asking...
For now, I'll play safe and answer: noon and midnight
Imagine a clock with hour and minute hands that move continuously. Suppose someone transposes the hands. The clock remains otherwise unchanged and continues to run. At what times of the day would this altered clock then show reasonable correlations of hands even if the times thus shown are wrong?
Wilmer: yes, if the hands coincide we have trivial solutions. But what about the other times where the positions of the transposed hands look reasonable even if the time displayed is wrong.
E.g., when transposing the hands at a time between 9:28 and 9:30 (assumed) the two hands take a position to each other as if it were a time between 5:45 und 5:55 (assumed). At this moment the clock looks reasonable for someone who doesnt have the correct time otherwise. A few moments later this will not be the case anymore. Then the transposed hands will show an impossible "time", and so on. Keep in mind that the transposed hands rotate with the wrong speed!