The aim of the present paper is to point out some connections between geometric control theory and the theory of behaviours. Specifically, given a behaviour, we analyse the set of all autonomous subbehaviours and relate those to some polynomial matrix completion problems. We use this to analyse the different module structures on some vectorial polynomial spaces. Subsequently, we apply this analysis to the study and the polynomial characterization of output nulling subspaces, controlled invariant subspaces and the natural, feedback induced, F [z]-module structures in these spaces. Toeplitz operators and Wiener-Hopf factorizations play an important role.