  We extend the duality between acyclic orientations and totally cyclic
orientations on planar graphs to dualities on graphs on orientable surfaces by
introducing boundary acyclic orientations and totally bi-walkable orientations.
In addition, we provide a reciprocity theorem connecting local tensions and
boundary acyclic orientations. Furthermore, we define the balanced flow
polynomial which is connected with tension polynomial by duality and with
totally bi-walkable orientations by reciprocity.
