  We study maps on the set of permutations of n generated by the R\'enyi-Foata
map intertwined with other dihedral symmetries (of a permutation considered as
a 0-1 matrix). Iterating these maps leads to dynamical systems that in some
cases exhibit interesting orbit structures, e.g., every orbit size being a
power of two, and homomesic statistics (ones which have the same average over
each orbit). In particular, the number of fixed points (aka 1-cycles) of a
permutation appears to be homomesic with respect to three of these maps, even
in one case where the orbit structures are far from nice. For the most
interesting such "Foatic" action, we give a heap analysis and recursive
structure that allows us to prove the fixed-point homomesy and orbit
properties, but two other cases remain conjectural.
