In this paper the notion of doubly invariant (DI) equilibrium is introduced. The concept extends controlled and robustly controlled invariance notions to the context of two-person dynamic games. Each player tries to keep the state in a region of state space independently of the actions of the rival player. The paper gives existence conditions, criteria and algorithms for the determination of DI equilibria of linear dynamic games in discrete time. Two examples illustrate the results. The first one is in the area of fault-tolerant controller synthesis. The second is an application to macroeconomics.