Check out the matrix R in this html page:
It say, the matrix was derived like this:
The reflection transformation can be decomposed for convenience into a translation to the origin, a rotation mapping the mirror into the XY plane, a scale of -1 in Z, the inverse of the rotation previously used, and a translation back to the mirror location.
Now to the the “mapping the mirror into the XY plane” part. I understand the new z axis in the mirror coordinate system is the normal of the mirror. But how about the x and y axes?