[en] Statistical invariance of Wiener increments under SO(n) rotations provides a notion of gauge transformation of state-dependent Brownian motion. We show that the stochastic dynamics of non-gauge-invariant systems is not unambiguously defined. They typically do not relax to equilibrium steady states even in the absence of external forces. Assuming both coordinate covariance and gauge invariance, we derive a second-order Langevin equation with state-dependent diffusion matrix and vanishing environmental forces. It differs from previous proposals but nevertheless incorporates the Einstein relation, a Maxwellian conditional steady state for the velocities, and the equipartition theorem. The overdamping limit leads to a stochastic differential equation in state space that cannot be interpreted as a pure differential (Itō, Stratonovich or other). At odds with the latter interpretations, the corresponding Fokker–Planck equation admits an equilibrium steady state; a detailed comparison with other theories of state-dependent diffusion is carried out. We propose this as a theory of diffusion in a heat bath with varying temperature. Besides equilibrium, a crucial experimental signature is the nonuniform steady spatial distribution.