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. In this paper 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.