 Coherent cyclic order of a digraph

A cyclic ordering of a directed graph is considered unchanged by rotations or swaps of consecutive nonadjacent vertices. A representative is coherent when every backward edge $v_j\to v_i$ has a forward directed path from $v_i$ to $v_j$. Equivalently every edge is on a directed circuit that winds exactly once around the order. The number of backward edges on a circuit is invariant under the allowed changes. Every nontrivial strongly connected directed graph admits such an order.