For reactions in which the initial and final states in the t channel contain equal-mass particles (e.g., NN̄-->ππ) of masses m and m', we show, using analytic continuation and Lorentz invariance, that the on-mass-shell helicity amplitudes in the region t=0, (m- m')2<=s<=(m+m')2 are invariant under the group O(4). Decompositions of the amplitudes in irreducible representations of O(4) (four-dimensional partial-wave expansions) are obtained and related to conventional partial-wave expansions. Poles classified according to the O(4) group are shown to lead to infinite families of Regge poles. The formalism is developed for arbitrary spins, and the case of nucleon-nucleon scattering is studied in detail. Our results for the Regge-pole structure in NN scattering are stronger than those of the conspirator theory.