  Since the work of Creutz, identifying the group identities for the Abelian
Sandpile Model (ASM) on a given lattice is a puzzling issue: on rectangular
portions of Z^2 complex quasi-self-similar structures arise. We study the ASM
on the square lattice, in different geometries, and a variant with directed
edges. Cylinders, through their extra symmetry, allow an easy determination of
the identity, which is a homogeneous function. The directed variant on square
geometry shows a remarkable exact structure, asymptotically self-similar.
