The reflection principle for Brownian motion roughly states that a Brownian motion reflected a stopping time is also a Brownian motion. More precisely, if $W$ is a Brownian motion and $T$ a stopping time then $\hat{W}_t = W_t$ on $t\leq T$ and $\hat{W}_t = 2W_T-W_t$ if $t>T$ is also a Brownian motion.
The proof of the reflection principle as far as I am aware only uses symmetry of the normal distribution, continuity of paths and the strong Markov property of $W$. My inclination is that the reflection would surely hold for a larger class of processes (possibly continuous symmetric stable processes?) than the Brownian motion, but I have yet to find a generalization.
My question is then if there are known generalizations of the reflection principle, or if not is it that continuous symmetric Markov processes are precisely Brownian motions?