Why do mirrors flip left to right and not up to down?

Mirrors do not flip left to right. They flip front to back. Mirrors reflect light rays such that the angle of reflection equals the angle of incidence.

For a three-dimensional object standing in front of the mirror, there is an image of the object created behind the mirror with the right side still on the right, the left side still on the left, the top side still on the top, the bottom side still on the bottom, but the front side is flipped to the back and the back side is flipped to the front.

This can be easily seen by tracing the rays from significant points on the object and seeing where they end up in the mirror image.