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We introduce two operators on stable configurations of the sandpile modelthat provide an algorithmic bijection between recurrent and parkingconfigurations. This bijection preserves their equivalence classes with respectto the sandpile group. The study of these operators in the special case of thecomplete bipartite graph ${K}_{m,n}$ naturally leads to a generalization of thewell known Cyclic Lemma of Dvoretsky and Motzkin, via pairs of periodicbi-infinite paths in the plane having slightly different slopes. We achieve ourresults by interpreting the action of these operators as an action on a pointin the grid $\mathbb{Z}^2$ which is pointed to by one of these pairs of paths.Our Cyclic lemma allows us to enumerate several classes of polyominoes, andtherefore builds on the work of Irving and Rattan (2009), Chapman et al.(2009), and Bonin et al. (2003).

Two operators on sandpile configurations, the sandpile model on the complete bipartite graph, and a Cyclic Lemma