Inner reflectors and non-orientable regular maps

Regular maps on non-orientable surfaces are considered with particular reference to the properties of inner reflectors, corresponding to symmetries of the 2-fold smooth orientable covering which project onto local reflections of the map itself. An example is given where no inner reflector is induced by an involution, and the existence of such involutions is related to questions of symmetry of coset diagrams for the symmetry group of the map.