We study the entanglement among helicity degrees of freedom in quantum electrodynamics (QED) scattering processes at fixed momentum, modeled as quantum maps whose spectral structure determines the associated entanglement dynamics. We show that, for fermion-to-fermion scatterings, parity invariance preserves maximal entanglement and, in an overwhelming number of cases, constraints the fixed points of the maps to be maximally entangled states. The structure of the maps also provides an accurate description of the entanglement dynamics in processes involving both fermions and photons.