  The symmetry properties which determine the critical exponents and
universality classes in conservative sandpile models are identified. This is
done by introducing a set of models, including all possible combinations of
abelian vs. non-abelian, deterministic vs. stochastic and isotropic vs.
anisotropic toppling rules. The universality classes are determined by an
extended set of critical exponents, scaling functions and geometrical features.
Two universality classes are clearly identified: (a) the universality class of
abelian models and (b) the universality class of stochastic models. In
addition, it is found that non-abelian models with deterministic toppling rules
exhibit non-universal behavior.
