We investigate a specific version of mirror symmetry for Z_M orbifold conformal field theories on K3. Our starting point is a description of mirror symmetry for toroidal models which is a simple precursor of the Strominger/Yau/Zaslow approach and which allows for an explicit formulation both in geometric and conformal field theoretic terms. Moreover, the Z_M orbifold construction can be carried out independently in both pictures and allows for a continuation of the mirror map to orbifold limits of K3 on the one hand and orbifold conformal field theories on the other. We determine explicit formulas for this mirror map as an element of the modular group of the moduli space of N=(4,4) superconformal field theories on K3 found by Aspinwall and Morrison. By comparing the geometric results to the conformal field theoretic ones we are able to determine the geometric counter parts of the twist fields in our orbifold conformal field theories.