It seems if I reverse velocities then things begin orbiting backwards, at least in classical mechanics.
Every orbit and trajectory outside atmospheres is in principle reversible, i.e., in the space-time function the time is reversed. The velocities are reversed and the accelerations are the same, including those due to rocket bursts. Thus if a rocket burst is in the direction of the velocity, in the reversed case it is opposite to the velocity. Of course in the case of rocket bursts there is no full reversal of events, both ways the same delta-v is used and the same mass ratio applies.
What's up when I put relativistic effects into the mix?
So for example I watch a super light test particle orbiting a black hole in a highly precessing flower shaped orbit. Then I put a bouncy wall into it's path that's at rest from my viewpoint when the particle hits it, so the particle bounces back reversing it's velocity from my viewpoint.
Would it begin running its orbits backwards?
The actual reason I'm asking this, because I want to know whether I can use backwards ray-tracing to render a black hole.