The only digits which appear the same when reflected are $0$, $1$, $3$ and $8$, so we want to find the number of times the display on the clock is made up of these digits.

The first digit can be $0$ or $1$, the second digit can be $0$, $1$, $3$ or $8$, the third digit can be $0$, $1$ or $3$ and the fourth digit can be $0$, $1$, $3$ or $8$. Therefore there are $2\times 4\times 3\times 4=96$ different possible displays that are the same when reflected.