Elementary particle models with internal degrees of freedom have been investigated within the framework of special relativity and orthodox quantum mechanics. Classical arguments indicate that systems whose extensions are \ensuremath{\lesssim} their Compton wavelength have spin excitation energies \ensuremath{\gtrsim} their rest mass. The principal aim of this paper is enumeration and classification of particles with rigid internal structure and a useful classification of particle models is by their symmetry groups. In nonrelativistic mechanics this classification shows that there are only the three well-known types of rigid systems that might be labeled by number of degrees of freedom as [0], [2], and [3] and are exemplified by an ideal point, diatomic molecule and rotator, respectively; while of the three types, but one, [3], possesses a spin-\textonehalf{} state of the Pauli-electron type. The corresponding analysis for relativistic mechanics shows there are nine types labeled here [0], [2], [3], [3\ensuremath{'}], [4], [4\ensuremath{'}], [4\ensuremath{''}], [5], and [6], and in addition two one-parameter infinities of types [${3}_{f}$] and [${5}_{f}$] ($0\ensuremath{\le}f\ensuremath{\le}\ensuremath{\pi}$). An algorithm exists for obtaining the spin-spectra of rigid structures from their symmetry groups. Of the $9+2\ensuremath{\infty}$ types, just three ([4], [5], and [6]) possess spin-\textonehalf{} states of the Dirac-electron type. The apparent rest mass depends upon the internal rotational state of the particle, as is shown by an unrealistic example of a Lagrangian which is an extension of that of the Klein-Gordon particle.