We consider discrete invertible dynamical systems L with the property that the kth iterate L^k possesses (reversing) symmetries mat are not possessed by L. A map U is called a (reversing) k-symmetry of L if k is the smallest positive integer for which U is a (reversing) symmetry of L^k. In this paper we discuss the particular case that L possesses a cyclic reversing k-symmetry group. We derive a decomposition property of maps that possess a cyclic reversing k-symmetry group and we classify the occurrence of such groups in invertible dynamical systems. We discuss the occurrence of nonsimultaneously linearizable nonisomorphic reversing k-symmetry groups in maps possessing cyclic reversing k-symmetry groups, illustrated by an example of a diffeomorphism on the plane R². We also construct examples of diffeomorphisms with cyclic reversing k-symmetry groups on the circle S¹, on the two-torus T², and on the cylinder S¹*R.