There exist classical systems whose canonical quantization yields relativistic wave equations. As a constructive proof, the classical mechanics of a translating-rotating five-frame is considered. Its quantization yields the Dirac, Weyl, Klein-Gordon, Maxwell-Proca, and higher-spin equations, together with a rotational mass spectrum for the states predicted. The motion of the free classical particle is studied, its anomalous behavior shedding light on the general breakdown of the probability interpretation of relativistic wave equations. Interactions are next discussed. Electromagnetic interactions, in Maxwell field and Fokker action-at-a-distance formulations, yield a theory that is the classical analog of quantum electrodynamics. Point-contact interactions between particles are contemplated, and it is shown that, at the classical level, it is impossible to formulate these in a Lorentz-invariant way. In addition, it is found that many-body systems are subject to constraint ideals, constituting irreducible modules under the action of the symmetric group. These features suggest that a correct quantum treatment of interactions may encounter unanticipated complexity. The conjectured interpretation of the theory, and its connection with gravitation, are briefly discussed. The paper concludes with a mathematical principle relating ordinary differential equations to families of linear partial differential equations, and this is further illustrated in the case of the Cauchy-Riemann equations of analytic function theory.