I discuss how to construct a Lorentz-invariant S-matrix for the scattering of electrically and magnetically charged particles. A key ingredient is a revision of our fundamental understanding of multi-particle representations of the Poincaré group. Surprisingly, the asymptotic states for electric-magnetic scattering transform with an additional little group phase, associated with pairs of electrically and magnetically charged particles. I will discuss the general construction of such states. The resulting "pairwise helicity" is identified with the quantized "cross product" of charges e1 g2- e2 g1 for every charge-monopole pair, and represents the extra angular momentum stored in the asymptotic electromagnetic field. We define a new kind ofpairwise spinor-helicity variable, which serves as an additional building block for electric-magnetic scattering amplitudes. We then construct the most general 3-point S-matrix elements, as well as the full partial wave decomposition for the 2 -> 2 fermion-monopole S-matrix. In particular, we derive the famous helicity flip in the lowest partial wave as a simple consequence of a generalized spin-helicity selection rule, as well as the full angular dependence for the higher partial waves. We will also discuss a possible direction for resolving Callan's ``semiton" problem: the scattering amplitude of a positron on a GUT monopole which apparently does not have an allowed final state. We will show that using entagled pairwise helicity spinors a simple possible s-wave final state does exist.