  We introduce and study the notion of continuous orbit equivalence of actions
of countable discrete groups on Cartan pairs in (twisted) groupoid context. We
characterize orbit equivalence of actions in terms of the corresponding
C$^*$-algebraic crossed products using Kumjian-Renault theory. We relate our
notion to the classical notion of orbit equivalence of actions on topological
spaces by showing that, under certain conditions, orbit equivalence of actions
on (twisted) groupoids follows from orbit equivalence of restricted actions on
unit spaces. We illustrate our results with concrete examples of continuous
orbit equivalent actions on groupoids coming from odometer transformations.
