Abstract: Regularity structures were introduced by Martin Hairer to deal with the divergences that arise from stochastic partial differential equations which typically involve white noise. Exploring the underlying geometry reveals the role played in this context by direct connections on a vector bundle. Originally introduced by Nikolai Teleman in the context of non commutative geometry, these provide a direct transport of fibres from point to point. We generalise them to groupoids and propose an interpretation of re-expansion maps arising in regularity structures in the language of groupoids. Re-expansion maps were introduced by Hairer to transform a singular stochastic differential equation into a fixed point problem, based on an ad hoc ``Taylor expansion'' of the solutions at any point in space-time and a ``re-expansion map'' which relates the values at different points. For gauge groupoids, namely those built from a principal bundle, a re-expansion map can be viewed as a (local) ``gaugeoid field'', the groupoid counterpart of a (local) gauge field. We investigate the case of jet bundles arising in polynomial regularity structures.