Tools from convex analysis are used to show how stability properties and Lyapunov inequalities translate when passing from a linear differential inclusion (LDI) to its dual. In particular, it is proved that a convex, positive definite function is a Lyapunov function for an LDI if and only if its convex conjugate is a Lyapunov function for the LDIs dual. Examples show how such duality effectively doubles the number of tools available for assessing stability of LDIs.