We generalize the concept of (reversing) symmetries of a dynamical sys
tem, i.e. we study dynamical systems that possess symmetry properties
only if considered on a proper time scale. In particular (considering
dynamical systems with discrete time), the kth iterate of a map may po
ssess more (reversing) symmetries than the map itself. In this way the
concepts of (reversing) symmetries and (reversing) symmetry groups ar
e generalized to (reversing) k-symmetries and (reversing) k-symmetry g
roups. Furthermore, a method is studied for finding orbits that are (k
-) symmetric with respect to reversing (k-)-symmetries. Firstly an exi
sting method for finding orbits that are symmetric with respect to one
reversing symmetry is extended to the case of more than one reversing
symmetry and secondly a generalization of this method to the case of
reversing k-symmetries is introduced. Some physically relevant example
s of dynamical systems possessing reversing k-symmetries are discussed
briefly.