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We make precise and prove a conjecture of Klivans about actions of thesandpile group on spanning trees. More specifically, the conjecture states thatthere exists a unique ``suitably nice'' sandpile torsor structure on planegraphs which is induced by rotor-routing.  First, we rigorously define a sandpile torsor algorithm (on plane graphs) tobe a map which associates each plane graph (i.e., planar graph with anappropriate ribbon structure) with a free transitive action of its sandpilegroup on its spanning trees. Then, we define a notion of consistency, whichrequires a torsor algorithm to be preserved with respect to a certain class ofcontractions and deletions. Using these definitions, we show that therotor-routing sandpile torsor algorithm is consistent. Furthermore, wedemonstrate that there are only three other consistent algorithms on planegraphs, which all have the same structure as rotor-routing.  We also define sandpile torsor algorithms on regular matroids and suggest anotion of consistency in this context. We conjecture that theBackman-Baker-Yuen algorithm is consistent, and that there are only three otherconsistent sandpile torsor algorithms on regular matroids, all with the samestructure.

Rotor-Routing Induces the Only Consistent Sandpile Torsor Structure on Plane Graphs