For two-dimensional periodic Kelvin-filtered Navier–Stokes systems, both positively and negatively invariant sets , consisting of initial data for which solutions exist for all negative times and exhibiting a certain asymptotic behaviour backwards in time, are investigated. They are proven to be rich in the sense that they project orthogonally onto the sets of lower modes corresponding to the first n distinct eigenvalues of the Stokes operator. In general, this yields the density in the phase space of trajectories of global solutions, but with respect to a weaker norm. This result applies equally to the two-dimensional periodic Navier–Stokes equations (NSEs) and the two-dimensional periodic Navier–Stokes-α model. We designate a subclass of filters for which the density follows in the strong topology induced by the (energy) norm of the phase space, as originally conjectured for the NSEs by Bardos and Tartar (1973 Arch. Ration. Mech. Anal. 50 10–25).