This work interprets and generalizes consensus-type algorithms as switching dynam-ics leading to symmetrization of some vector variables with respect to the actions of a finite group.We show how the symmetrization framework we develop covers applications as diverse as consensuson probability distributions (either classical or quantum), uniform random state generation, andopen-loop disturbance rejection by quantum dynamical decoupling. Robust convergence results areexplicitly provided in a group-theoretic formulation, both for deterministic and for randomized dy-namics. This indicates a way to directly extend the robustness and randomization properties ofconsensus-type algorithms to more fields of application.