Within the framework of relativistic Schrödinger theory (RST), the problem of bound two-particle states is studied and compared to the analogous results of conventional quantum theory. The standard dichotomy of symmetric and antisymmetric quantum states finds its RST analog in the form of the positive and negative mixtures. Similarly the conventional exchange degeneracy has its RST counterpart in the form of a certain mixture degeneracy which, however, is not broken by the interparticle interactions as in the standard quantum theory. The corresponding group of quasilinear mixing transformations turns out to be continuous [in contrast to the discrete operations of (anti)symmetrization in the conventional theory] and is closely related to the Lorentz group SO(1,1) in two dimensions. The group properties can be exploited to generate exact solutions for the mixture dynamics from the pure-state configurations where, however, certain physical quantities remain invariant: scalar densities, currents, energy eigenvalues.