D. Wilczak, P. Zgliczynski Topological method for symmetric periodic orbits for maps with a reversing symmetry (122K, LaTeX 2e with 4 PS Figures) ABSTRACT. We present a topological method of obtaining the existence of infinite number of symmetric periodic orbits for systems with reversing symmetry. The method is based on covering relations. We apply the method to a four-dimensional reversible map.