Let f be a polynomial automorphism of the affine plane. In this paper we consider the possibility for it to possess infinitely many periodic points on an algebraic curve C. We conjecture that this hap- pens if and only if f admits a time-reversal symmetry; in particular the Jacobian Jac(f) must be a root of unity. As a step towards this conjecture, we prove that its Jacobian, together with all its Galois conjugates lie on the unit circle in the complex plane. Under mild additional assumptions we are able to conclude that indeed Jac(f) is a root of unity. We use these results to show in various cases that any two automor- phisms sharing an infinite set of periodic points must have a common it- erate, in the spirit of recent results by Baker-DeMarco and Yuan-Zhang.