  Interval-closed sets of a poset are a natural superset of order ideals. We
initiate the study of interval-closed sets of finite posets from enumerative
and dynamical perspectives. In particular, we use the generalized toggle group
to define rowmotion on interval-closed sets as a product of these toggles. Our
main theorem is an intricate global characterization of rowmotion on
interval-closed sets, which we show is equivalent to the toggling definition.
We also study specific posets; we enumerate interval-closed sets of ordinal
sums of antichains, completely describe their rowmotion orbits, and prove a
homomesy result involving the signed cardinality statistic. Finally, we study
interval-closed sets of product of chains posets, proving further results about
enumeration and homomesy.
