Rigid transformations are involved in a wide range of digital image processing applications. In such a context, they are generally considered as continuous processes, followed by a digitization of the results. Recently, rigid transformations on $\Zset^2$ have been alternatively formulated as a fully discrete process. Following this paradigm, we investigate --from a combinatorial point of view-- the effects of pixel-invariance constraints on such transformations. In particular we describe the impact of these constraints on both the combinatorial structure of the transformation space and the algorithm leading to its generation.