  On-shell methods are particularly suited for exploring the scattering of
electrically and magnetically charged objects, for which there is no local and
Lorentz invariant Lagrangian description. In this paper we show how to
construct a Lorentz-invariant S-matrix for the scattering of electrically and
magnetically charged particles, without ever having to refer to a Dirac string.
A key ingredient is a revision of our fundamental understanding of
multi-particle representations of the Poincar\'e 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. The corresponding "pairwise helicity" is identified with the
quantized "cross product" of charges, $e_1 g_2 - e_2 g_1$, for every
charge-monopole pair, and represents the extra angular momentum stored in the
asymptotic electromagnetic field. We define a new kind of pairwise
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 $2\to 2$ 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. Our construction provides a
significant new achievement for the on-shell program, succeeding where the
Lagrangian description has so far failed.
