The combined problem of disturbance rejection and decoupling for linear systems is considered. New necessary and sufficient conditions are derived for the existence of a state feedback controller that rejects the disturbances and simultaneously decouples the closed-loop system. These conditions are simple, intuitively appealing and are expressed in terms of system structural invariants such as infinite zero orders. Then we consider the disturbance rejection with simultaneous decoupling problem for structured linear systems. Necessary and sufficient conditions for the generic solvability of this problem are provided. They are expressed in terms of input-output path length comparisons on the system associated graph.