The cluster modular group of the dimer model

Associated to a convex integral polygon $P$ is a dimer cluster integrable system $\mathcal X_P$ of Goncharov and Kenyon. We compute the cluster modular group $G_P$ of $\mathcal X_P$. Probabilistically, non-torsion elements of $G_P$ correspond to ways of shuffling the underlying bipartite graph, generalizing the domino shuffling algorithm.