We consider decay chains of the type P -> M + ... -> f + ..., where M is a neutral meson that may mix with its antiparticle Mbar, before decaying into the final state f. P may be either a heavier neutral meson or a charged meson. We perform a rephasing-invariant analysis of the quantities that show up in such cascade decays. If the decay P -> M + ... (or the decay P -> Mbar + ...) is forbidden, we find the usual lambda_f parameters describing the interference between the mixing of a neutral meson system and the decay from that system into the final state f. However, when both the P -> M + ... and P -> Mbar + ... decays are allowed, we find a new class of rephasing-invariant parameters, xi_i, that measure the interference between the mixing of a neutral meson system and the decay from the initial state *into that system*. We show that the quantities lambda_f and xi_i are necessary and sufficient to describe all the interference effects present in the most general cascade decay. We discuss the various cascade decays in turn, highlighting the special features of each one. Comment: 22 pages, latex, 5 figures, corrected missing signs in the discussion of the CP conservation limit