  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 happens if and only if $f$
admits a time-reversal symmetry; in particular the Jacobian $\mathrm{Jac}(f)$
must be a root of unity.
  As a step towards this conjecture, we prove that the Jacobian of $f$ and all
its Galois conjugates lie on the unit circle in the complex plane. Under mild
additional assumptions we are able to conclude that indeed $\mathrm{Jac}(f)$ is
a root of unity. We use these results to show in various cases that any two
automorphisms sharing an infinite set of periodic points must have a common
iterate, in the spirit of recent results by Baker-DeMarco and Yuan-Zhang.
