  Inspired by a twist maps theorem of Mather we study recurrent invariant sets
that are ordered like rigid rotation under the action of the lift of a bimodal
circle map $g$ to the $k$-fold cover. For each irrational in the interior of
the rotation set the collection of the $k$-fold ordered semi-Denjoy minimal
sets with that rotation number contains a $(k-1)$-dimensional ball in the weak
topology on their unique invariant measures. We also describe completely their
periodic orbit analogs for rational rotation numbers. The main tool is a
generalization of a construction of Hedlund and Morse which generates the
symbolic analogs of these $k$-fold well-ordered invariant sets.
