The parameterizations of rotation and motion are the subject of continuous research and development in many theoretical and applied fields of mechanics such as rigid body, structural, and multibody dynamics. Tensor analysis expresses the invariance of the laws of physics with respect to the change of basis and change of frame operations. Consequently, it is imperative to formulate mechanics problems in terms of tensorial quantities. This paper presents formal proofs that rotation and motion parameterizations are tensors if and only if they are parallel to the eigenvectors of the rotation and motion tensors, respectively, associated with their unit eigenvalues. Furthermore, it also establishes that the tangent operators of rotation and motion are of a tensorial nature if and only if they are expressed in terms of the vectorial parameterizations of rotation and motion, respectively. Finally, important tensor identities are shown to hold for all vectorial parameterization of rotation and motion.