When I saw the mirrors in Duke3D for the first time I wondered how they pulled that off. I had made a couple of Wolfenstein like tech demos at the time, going off the published stuff about the algorithms and techniques used by Carmack and Romero at the time (groundbreaking stuff). So I delved into it at the time and the way they did it was to have an identical but mirrored room behind the mirror. The mirror was just a transparent wall. And they had a sprite spawn in that looked like Duke and was controlled by the player.
Mirrors in games are hard.