Orthogonal Mueller matrices can be considered as corresponding either to retarders or to generalized transformations of the polarization basis for the representation of Stokes vectors, so that they constitute the only type of Mueller matrices that preserve the degree of polarization and the intensity of any partially polarized input Stokes vector. The physical quantities that remain invariant when a nondepolarizing Mueller matrix is transformed through its product by different types of orthogonal Mueller matrices are identified and interpreted, providing a better knowledge of the information contained in a nondepolarizing Mueller matrix.