Homomesy is an invariance phenomenon in dynamical algebraic combinatorics which
occurs when the average value of some statistic on a set of combinatorial objects is the
same over each orbit generated by a map on these objects. We perform a systematic search
for statistics homomesic for the set of permutations under the rotation map, identifying
and proving 34 instances of homomesy. We show that these homomesies actually hold
not only for rotation but in fact for a whole class of maps related to rotation by the notion
of toggling, which is identified initially with composition of simple transpositions. In
this way these maps are related to the rowmotion action defined on various combinatorial
structures, which has a useful definition in terms of toggling. We prove some initial results
on maps given by restricted or modified toggles. We discuss also the computational method
used to identify candidate statistics from FindStat, a combinatorial statistics database.
