The difference between the two times is 555 minutes, which is 9 hours 15 minutes.
Looking only at the minute hand, there must therefore be a 15 or 45 minute difference between them. The minute hand difference between X past _ and Y to _ is X+Y, so X+Y=15 or X+Y=45. The latter would lead to X>12 or Y>12 which is not possible since they should also represent one of the hours of a clock.
This is all the question asks for, but we can actually work out the time completely.
Now look at the whole hours. There is a difference of 9 hours between them, so you would expect |Y-X|=9 or |Y-X|=3. I ignored the minutes here, and doing so could make the real difference one hour more or less. However, from the previous equation the sum X+Y is odd, so the difference |Y-X| must also be odd, and cannot be off by one.
Solving the two equations, we get 4 potential solutions:
This gives the times:
3 to 3, and 12 past 12: 2:57, 12:12
12 to 12, and 3 past 3: 11:48, 3:03
6 to 6, and 9 past 9: 5:54, 9:09
9 to 9, and 6 past 6: 8:51, 6:06
The second and third of these don't have a difference of 9:15 but of 8:45, so are invalid.
1000 minutes is equal to 16 hours and 40 minutes, or +4:40. The two potential solutions are then
Y to Y X past X After 1000 minutes
2:57 12:12 : 7:37 4:52
8:51 6:06 : 1:31 10:46
The second of these can be considered to have a difference of 9:15 without adding/subtracting 12 hours. So the best solution is:
9 to 9 and 6 past 6
Nevertheless, the solution 3 to 3 and 12 past 12 is valid too if you allow the difference to span across the 12 hour mark.