  We introduce symplectic billiards for pairs of possibly non-convex polygons.
After establishing basic properties, we give several criteria on pairs of
polygons for the symplectic billiard map to be fully periodic, i.e.
$\textit{every}$ orbit is periodic. First fully periodic examples were
discovered by Albers-Tabachnikov [AT18] and
Albers-Banhatti-Sadlo-Schwartz-Tabachnikov in [ABS+19]. Our criteria allow us
to construct a plethora of new examples. Moreover, we provide an example of a
pair of polygons where the symplectic billiard map is fully periodic while
having orbits of arbitrarily large period. After giving a class of examples
which provably have isolated periodic orbits (and are thus not fully periodic)
we exhibit the first example without any periodic orbits at all. It is open
whether being fully periodic with unbounded period or having no periodic orbits
at all is possible in the single polygon setting. Finally, we prove that if one
replaces polygons by smooth strictly convex curves then there are always
infinitely many periodic orbits.
