We develop a gauge-invariant formalism suitable for the study of cosmological perturbations induced by defects. It consists in remolding the well-known Bardeen formalism into a local and gauge-invariant theory formulated in configuration space. Bardeen's variables are first combined into local and gauge-invariant variables with simple geometric interpretations, and a system of hyperbolic equations is then derived for the new variables. The resulting theory is manifestly causal and it contains the density-gradient-vorticity interaction effect first discovered by Ellis, Bruni, and Hwang