  This paper analyzes a certain action called "whirling" that can be defined on
any family of functions between two finite sets equipped with a linear (or
cyclic) ordering. As a map on injections and surjections, we prove that within
any whirling-orbit, any two elements of the codomain appear as outputs of
functions the same number of times. This result, can be stated in terms of the
homomesy phenomenon, which occurs when a statistic has the same average across
every orbit. We further explore whirling on parking functions, order-preserving
maps, and restricted growth words, discussing homomesy results for each case.
