This work deals with compensation of inaccessible or measurable disturbances in a class of switching systems: i.e., those defined by a set of discrete-time linear time-invariant modes and a switching signal satisfying a minimum dwell-time constraint. In more detail, the problem consists in finding a mode-dependent, switching compensation scheme that guarantees zero output asymptotically, for any admissible disturbance, any initial state of the compensated switching system, and any admissible switching signal. The proposed switching compensator performs both a feedforward action and a feedback action, in the case of measurable signals - which, in this sense, is regarded as the more general one. Necessary and sufficient conditions for the existence of such compensator are proven. On those conditions, a complete synthesis procedure is presented. A numerical example is worked out, with the twofold objective of showing how to apply the computational algorithms for the synthesis and illustrating the effectiveness of the devised compensation scheme.