  We define the analogue of linear equivalence of graph divisors for the
rotor-router model, and use it to prove polynomial time computability of some
problems related to rotor-routing. Using the connection between linear
equivalence for chip-firing and for rotor-routing, we give a simple proof for
the fact that the number of rotor-router unicycle-orbits equals the order of
the Picard group. We also show that the rotor-router action of the Picard group
on the set of spanning in-arborescences can be interpreted in terms of the
linear equivalence.
