This note introduces continuous-time linear systems with switching parameters driven by a Markov chain running in reverse time. Reversibility of the chain is not imposed, so the system cannot be recast as standard Markov jump linear systems (MJLS), demanding for a specific study. We develop the operator-based approach, which is elaborate in view of possible multiplicity of solutions for the second moment variable, as detailed in this note. We also study the optimal quadratic control problem based on Hamilton-Jacobi-Bellman equations and standard dynamic programming. The control solution is given in terms of a coupled Riccati equation that is harmonious with filtering of MJLS, which allows us to generalize the classic control-filtering duality for linear time varying systems.