Relativity theory permits motions of a free-spinning particle in which the instantaneous velocity and linear momentum are not collinear. Therefore, it is necessary to specify two invariance groups in order to completely describe the space-time symmetry properties of a particle with intrinsic spin. If the quantal description of such a particle is given by a covariant linear field equation, then the external space-time symmetry is specified by the 10-element Poincar\'e group generated by the linear four-momentum and total angular momentum operators. However, invariance of the field equation under external inhomogeneous Lorentz transformations does not complete the algebra of the 10-element group generated by the instantaneous four-velocity and spin angular-momentum operators. Several formulations of linear field equations admitting a mass-spin spectrum are based on different choices for this latter, intrinsic, space-time symmetry group. We begin with the simplest choice, namely, intrinsic Poincar\'e invariance, and establish the formal connection between these several formulations by constructing a unitary transformation which generates an infinite sequence of linear wave equations describing ascendingly more complex intrinsic space-time symmetry, but linked by a common mass operator. Of special interest is the infinitesimal transformation which generates the familiar intrinsic DeSitter group. Finally, some kinematical properties of this transformation are discussed for various proposed mass operators.