  Electric-magnetic duality, the Newman-Janis shift, and the double copy all
act by elementary operations on three-point amplitudes. At the same time, they
generate a network of interesting classical solutions spanning from the Coulomb
charge via the dyon to the Kerr-Taub-NUT spacetime. We identify the amplitudes
corresponding to each of these solutions, working to all orders in spin, but to
leading perturbative order. We confirm that the amplitudes double-copy when the
solutions are related by the classical double copy. Along the way we show that
the Kerr-Taub-NUT solution corresponds to a gravitational electric-magnetic
duality rotation acting on the Kerr solution, again to all orders in spin, and
demonstrate that the asymptotic charges also transform simply under our
operations.
