We analyze large gauge invariance in the non-Abelian, finite temperature context and its physical consequences for D=3 effective actions. After briefly reviewing the structure of bundles and large gauge transformations that arise in nonsimply connected 3-manifolds and gauge groups, we discuss their connections to Chern-Simons terms and Wilson-Polyakov loops. We then provide an invariant characterization of the “Abelian” fluxes encountered in explicit computations of finite temperature effective actions. In particular we relate, and provide explicit realizations of, these fluxes to a topological index that measures the obstruction to global diagonalization of the loops around compactified time. We also explore the fate of, and exhibit some everywhere smooth, large transformations for nonvanishing index in various topologies.