A Mirror Reverses One Axis, and It Is Not Left or Right
Raise your right hand at a mirror and the reflected hand stays on the same side of the room, which means the usual question has a false premise. A plane mirror is the matrix diag(1, 1, -1): it fixes both axes lying in the glass and reverses only the direction you look along. Its determinant is -1, so no rotation reproduces it, and the sideways flip everyone reports belongs to the half turn you perform in your head.