  The abelian sandpile models feature a finite abelian group $G$ generated by
the operators corresponding to particle addition at various sites. We study the
canonical decomposition of $G$ as a product of cyclic groups $G = Z_{d_1}
\times Z_{d_2} \times Z_{d_3} >... \times Z_{d_g}$ where $g$ is the least
number of generators of $G$, and $d_i$ is a multiple of $d_{i+1}$. The
structure of $G$ is determined in terms of the toppling matrix $\Delta$. We
construct scalar functions, linear in height variables of the pile, that are
invariant under toppling at any site. These invariants provide convenient
coordinates to label the recurrent configurations of the sandpile. For an $L
\times L$ square lattice, we show that $g = L$. In this case, we observe that
the system has nontrivial symmetries, transcending the obvious symmetries of
the square, viz. those coming from the action of the cyclotomic Galois group
Gal$_L$ of the $2(L+1)$--th roots of unity (which operates on the set of
eigenvalues of $\Delta$). We use Gal$_L$ to define other simpler, though
under-complete, sets of toppling invariants.
