  The rowmotion operator acting on the set of order ideals of a finite poset
has been the focus of a significant amount of recent research. One of the major
goals has been to exhibit homomesies: statistics that have the same average
along every orbit of the action. We systematize a technique for proving that
various statistics of interest are homomesic by writing these statistics as
linear combinations of "toggleability statistics" (originally introduced by
Striker) plus a constant. We show that this technique recaptures most of the
known homomesies for the posets on which rowmotion has been most studied. We
also show that the technique continues to work in modified contexts. For
instance, this technique also yields homomesies for the piecewise-linear and
birational extensions of rowmotion; furthermore, we introduce a $q$-analogue of
rowmotion and show that the technique yields homomesies for "$q$-rowmotion" as
well.
