  Using on-shell amplitude methods, we explore 4-dimensional Electric-Magnetic
duality and its double copy. We show explicitly that the on-shell scattering
amplitudes know about `dual' photons (and dual gravitons), that the off-shell
photon propagator double copies to the graviton propagator and that the
magnetic part of the propagator is essential for the double copy to hold. We
also show that there is an equivalent gravito-magnetic part of the graviton
propagator which is essential in giving rise to solutions with either angular
momentum or NUT charge. Furthermore, we comment on the so-called Weinberg
paradox, which states that scattering amplitudes involving the mixing of
electric and magnetic monopoles cannot be Lorentz invariant, and would seem to
preclude the existence of the 't Hooft-Polyakov (topological) monopole. We
trace this paradox to the magnetic part of the propagator, showing that it can
be eliminated if one restricts to proper orthochronous Lorentz transformations.
Finally, we compute the fully relativistic cross-section for arbitrary spin
dyons using the recently formulated on-shell duality transformation and show
that this is always fully Lorentz invariant.
