  Cascades are self-reinforcing processes underlying the systemic risk of many
complex systems. Understanding the universal aspects of these phenomena is of
fundamental interest, yet typically bound to numerical observations in ad-hoc
models and limited insights. Here, we develop a unifying approach and show that
cascades induced by a long-range propagation of local perturbations are
characterized by two universality classes determined by the parity invariance
of the underlying process. We provide hyperscaling arguments predicting hybrid
critical exponents given by a combination of both mean-field spinodal exponents
and $d$-dimensional corrections and we show how global symmetries influence the
geometry and lifetime of avalanches. Simulations encompassing classic and novel
cascade models validate our predictions, revealing fundamental principles of
cascade phenomena amenable to experimental validation.
