  Let G be a connected, loopless multigraph. The sandpile group of G is a
finite abelian group associated to G whose order is equal to the number of
spanning trees in G. Holroyd et al. used a dynamical process on graphs called
rotor-routing to define a simply transitive action of the sandpile group of G
on its set of spanning trees. Their definition depends on two pieces of
auxiliary data: a choice of a ribbon graph structure on G, and a choice of a
root vertex. Chan, Church, and Grochow showed that if G is a planar ribbon
graph, it has a canonical rotor-routing action associated to it, i.e., the
rotor-routing action is actually independent of the choice of root vertex.
  It is well-known that the spanning trees of a planar graph G are in canonical
bijection with those of its planar dual G*, and furthermore that the sandpile
groups of G and G* are isomorphic. Thus, one can ask: are the two rotor-routing
actions, of the sandpile group of G on its spanning trees, and of the sandpile
group of G* on its spanning trees, compatible under plane duality? In this
paper, we give an affirmative answer to this question, which had been
conjectured by Baker.
