Today I just learned about the uncaught third strike rule in baseball:

On an uncaught third strike with (1) no runner on first base, or (2) with a runner on first base and two outs, the batter immediately becomes a runner. The strike is called, but the umpire does not call the batter out. The umpire may also signal that there is "no catch" of the pitch. The batter may then attempt to reach first base and must be tagged or forced out.

Honorary mention to the defensive possession after a touchdown in American football. I can't find it in the rulebook right now, but the essential concept is that, after a score, the team that was scored on actually has a choice on whether to receive the ball and start the next possession on offense (which is normally what happens), or whether to kick off the ball and defend (considering that the team that just scored on you gets the ball back to do it again, I don't see how this could ever be a good idea in today's game).