We prove that if $F$ is an orientation-preserving homeomorphism of the planethat leaves invariant an irreducible plane separating continuum $\Delta$,then, with the possible exception of three numbers, if $p/q$ is a reducedrational in the interior of the convex hull of the rotation set of$F\vert_{\Delta}$ (with respect to some lift) there are at least two distinctperiodic orbits of $F\vert_{\Delta}$ of period $q$ and rotation number $p/q$.This result also applies to certain nonseparating invariantcontinua.