  The Abelian Sandpile Model is a cellular automaton whose discrete dynamics
reaches an out-of-equilibrium steady state resembling avalanches in piles of
sand. The fundamental moves defining the dynamics are encoded by the toppling
rules. The transition monoid corresponding to this dynamics in the set of
stable configurations is abelian, a property which seems at the basis of our
understanding of the model. By including also antitoppling rules, we introduce
and investigate a larger monoid, which is not abelian anymore. We prove a
number of algebraic properties of this monoid, and describe their practical
implications on the emerging structures of the model.
