Discrete linear transformations are important tools in information processing. Many such transforms are injective and therefore prime candidates for a physically reversible implementation into hardware. We present here reversible digital implementations of different integer transformations on four inputs. The resulting reversible circuit is able to perform both the forward transform and the inverse transform. Which of the two computations that actually is performed, simply depends on the orientation of the circuit when it is inserted in a computer board (if one takes care to provide the encapsulation of symmetrical power supplies). Our analysis indicates that the detailed structure of such a reversible design strongly depends on the prime factors of the determinant of the transform: a determinant equal to a power of 2 leads to an efficient garbage-free design.