We present a systematic study of the dyon-dyon classical interaction in the framework of predictive relativistic mechanics. A uniquely defined expression for the accelerations of the system is obtained up to first order in the coupling constants. We analyze the symmetry properties of the system under discrete symmetries and obtain explicit expressions for Pμ and Jλμ at this order. Futhermore, we find a family of bound states with circular orbits that seems to be a generalization of the well-known exact solution of Schild. The scattering problem is also treated and second-order expressions for the cross sections are given which agree, by taking the appropriate limits, with the one previously found by Barut and Beker and also with the one obtained by Bel in the framework of a quantized version of predictive relativistic mechanics for spinless particles.