Rigid motions in $\mathbb{R}^2$ are fundamental operations in 2D image processing. They satisfy many properties: in particular, they are isometric and therefore bijective. Digitized rigid motions, however, lose these two properties. To investigate the lack of injectivity or surjectivity and more generally their local behavior, we extend the framework initially proposed by Nouvel and R\'emila to the case of digitized rigid motions. Yet, for practical applications, the relevant information is not global bijectivity, which is seldom achieved, but bijectivity of the motion restricted to a given finite subset of $\mathbb{Z}^2$. We propose two algorithms testing that condition. Finally, because rotation angles are rarely given with infinite precision, we propose a third algorithm providing optimal angle intervals that preserve this restricted bijectivity.