We show that every correctable subsystem for an arbitrary noise operation can be recovered by a unitary operation, where the notion of recovery is more relaxed than the notion of correction insofar as it does not protect the subsystem from subsequent iterations of the noise. As an application, we demonstrate that the noiseless subsystems for the composition of a unital quantum operation with its dual are precisely the correctable subsystems for the map that can be corrected by a single unitary operation. Using the recently developed structure theory for noiseless subsystems, the identification of such correctable subsystems is reduced to an algebraic exercise.