We study bisimulation-based information flow security properties which are persistent, in the sense that if a system is secure, then all states reachable from it are secure too. We show that such properties can be characterized in terms of bisimulation-like equivalence relations between the system and the system itself prevented from performing confidential actions. Moreover, we provide a characterization of such properties in terms of unwinding conditions which demand properties of individual actions. These two different characterizations naturally lead to efficient methods for the verification and construction of secure systems. We also prove several compositionality results and discuss a sufficient condition to define refinement operators preserving security.