Abstract: The Grassmannian admits an automorphism of finite order, the cyclic shift map. This map has finitely many fixed points, which were described by Steven Karp in a recent paper. We study the fixed-point set of any iterate of the cyclic shift map. These cyclic symmetry loci are typically positive-dimensional spaces. We give a simple geometric description of these loci, and of their totally nonnegative part. We describe an atlas of generalized cluster algebra charts on cyclic symmetry loci, whose clusters are efficient total positivity tests. These cluster algebras have connections with higher Teichmuller theory and with the category of representations of quantum affine algebras at roots of unity. Please email Colleen at cer2 (at) illinois (dot) edu for the Zoom ID and password.