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The study of obstacle scattering for the Klein-Gordon equation in thepresence of long-range magnetic potentials is addressed. Previous results ofthe authors are extended to the long-range case and the results the authorspreviously proved for high-momenta long-range scattering for the Schr\"odingerequation are brought to the relativistic scenario. It is shown that there areimportant differences between relativistic and non-relativistic scatteringconcerning long-range. In particular, it is proved that the electric potentialcan be recovered without assuming the knowledge of the long-range part of themagnetic potential, which has to be supposed in the non-relativistic case. Theelectric potential and the magnetic field are recovered from the high momentalimit of the scattering operator, as well as fluxes modulo $2 \pi $ aroundhandles of the obstacle. Moreover, it is proved that, for every $\hat{\mathbfv} \in \mathbb{S}^2$, $ A_\infty(\hat{\mathbf v}) + A_\infty(-\hat{\mathbf v})$can be reconstructed, where $A_\infty$ is the long-range part of the magneticpotential. A a simple formula for the high momenta limit of the scatteringoperator is given, in terms of magnetic fluxes over handles of the obstacle andlong-range magnetic fluxes at infinity, that are introduced in this paper. Theappearance of these long-range magnetic fluxes is a new effect in scatteringtheory.

High-Momenta Estimates for the Klein-Gordon Equation: Long-Range Magnetic Potentials and Time-Dependent Inverse Scattering