Let G be a graph and let C be a set of coloring. Suppose that there is an involution $\phi$ from C to C. We can think about the element of C as the nonzero elements of some Abelian group and $\phi(x)=-x$. (For concreteness we can consider especially the special case that C corresponds to the cyclic group, so that $\phi$ has no fixed points when |C| is even and one fixed point when |C| is odd.)

Consider a colorings where for every edge {u,v} in G we color the directed edge from u to v with some color c and the directed edge from v to u with -c. We will also require that two edges with the same tail and two edges with the same head must have different colors. If $\phi$ is the identity this is a usual edge coloring of $G$.

My question is if such colorings were considered and what is known about them. For example, is there an analog of Vizing theorem? (Vizing theorem asserts that the edges of every graph with maximum degree d can be colored by d+1 colors.)