Imagine moving parallel to the surface of a very large flat mirror at relativistic speeds. What would be the effect of viewing yourself in the mirror? At non-relativistic speeds your image would be similar to what you would see when stationary. Assume you are moving to your right with respect to the mirror at high speeds. Without doing the math I would expect your image would start moving to your left. Also, your image would start to distort because the path lengths of the photons from various parts of your body would take different times to arrive back at your eye. So what arrives simultanously to your eye would be different regions of yourself at different times in the past. Also, assuming that the light source that is illuminating you is moving with you, the photons would transfer some of their momentum and energy to the mirror, and so the light from the image would be red-shifted. The last part I am not sure about, is reflection a process of absorption and re-emission from the surface atoms of a mirror?
Remember that the situation you describe is exactly the same as you remaining stationary while the mirror moves past you at relativistic speeds. The reflection from a moving mirror is analysed in this article. If the mirror is parallel to it's direction of motion ($\phi$ = 0) the normal rules apply i.e. the angle of incidence is equal to the angle of reflection. So assuming the mirror is perfect, your reflection would appear to you to be unaffected. Observers in the mirror's rest frame would see something different of course. I would guess they will see you rotated, just as if they were looking at you directly.
I made the qualification "assuming the mirror is perfect" because as far as the mirror is concerned the light from you is arriving at a very shallow angle of order 1/$c$ radians (but still at the speed of light of course). I'm not sure whether there would be any physical effects that change the reflectivity at such shallow angles of incidence.
Plot the imaginary copy of yourself on the other side of the mirror, by the laws of geometrical optics, for every moment of time. *)
Then consider that copy a real body you observe, and apply all known relativistic effects to its apparent image, taking into account both your and its motion. In the case of a parralel flat mirror and uniform motion, all those effects would apply twice, for the motion of the body and for the motion of the observer, and accurately cancel each other, so you would see just your unchanged image. You can even wave your hand to it.
*) Footnote: For the moving mirror, you should plot your copy by the rule that it is reflected by the mirror surface taken in the later moment of time: take your position and the mirror's motion and find when photons from you would reach the mirror. The copy would not be shifted in time. Also, the mirror is considered infinitely heavier that the photons.