  For any link and for any modulus $m$ we introduce an equivalence relation on
the set of non-trivial m-colorings of the link (an m-coloring has values in
Z/mZ). Given a diagram of the link, the equivalence class of a non-trivial
m-coloring is formed by each assignment of colors to the arcs of the diagram
that is obtained from the former coloring by a permutation of the colors in the
arcs which preserves the coloring condition at each crossing. This requirement
implies topological invariance of the equivalence classes. We show that for a
prime modulus the number of equivalence classes depends on the modulus and on
the rank of the coloring matrix (with respect to this modulus).
