We have observed remarkable changes in the magnetic properties of excitons as they acquire kinetic energy. In particular, the Zeeman splittings and diamagnetic shifts of excitonic transitions when magnetic fields are applied along the growth direction of (001) wide quantum wells of CdTe, ZnSe, ZnTe and GaAs are found to to have a strong dependence on the translational wavevector Kz. The behaviour of the Zee-man splittings corresponds to enhancement of the magnetic moments of the excitons. This enhancement is particularly marked when their translational kinetic energy becomes comparable with the exciton Rydberg and can be described by what appears to be a universal function of Kz. A model for the behaviour is outlined which involves motionally-induced mixing between the 1S hydrogenic exciton ground state and excited nP states. The observations imply that there are significant changes in the structure of the exciton as its translational kinetic energy increases.