We formulate a two-team linear quadratic stochastic dynamic game featuring two opposing teams each with decentralized information structures. We introduce the concept of mutual quadratic invariance (MQI), which, analogously to quadratic invariance in (single team) decentralized control, defines a class of interacting information structures for the two teams under which optimal feedback control strategies are linear and easy to compute. We show that for zero-sum two-team dynamic games with MQI information structure, structured state feedback saddle-point equilibrium strategies can be computed from equivalent structured disturbance feedforward saddle point equilibrium strategies. We also show that there is a saddle-point equilibrium in linear strategies even when the teams are allowed to use nonlinear strategies. However, for nonzero-sum games we show via a counter-example that a similar equivalence fails to hold. The results are illustrated with a simple yet rich numerical example that illustrates the importance of the information structure for dynamic games.