  The Grassmannian admits an action by a finite cyclic group via the cyclic
shift map. We give a simple description of the points fixed by each element of
this cyclic group, extending Karp's description of the points fixed by the
cyclic shift itself. We give a cell decomposition of the set of totally
nonnegative points in each cyclic symmetry locus and describe efficient total
positivity tests, extending results of Postnikov to the cyclically symmetric
setting. We describe a conjectural generalized cluster structure on cyclic
symmetry loci provided the order of the orbifold point is sufficiently large.
The generalized exchange relations we find should be a Higher Teichm\"uller
analogue of the relations Chekhov and Shapiro used to study Teichm\"uller
theory of orbifolds.
