This paper discusses the possibility of using Compton scattering--an inelastic X-ray scattering process that yields a projection of the electron momentum density--to probe magnetoelectrical properties. It is shown that an antisymmetric component of the momentum density is a unique fingerprint of such time- and parity-odd physics. It is argued that polar ferromagnets are ideal candidates to demonstrate this phenomenon and the first experimental results are shown, on a single-domain crystal of GaFeO3. The measured antisymmetric Compton profile is very small (≃ 10(-5) of the symmetric part) and of the same order of magnitude as the statistical errors. Relativistic first-principles simulations of the antisymmetric Compton profile are presented and it is shown that, while the effect is indeed predicted by theory, and scales with the size of the valence spin-orbit interaction, its magnitude is significantly overestimated. The paper outlines some important constraints on the properties of the antisymmetric Compton profile arising from the underlying crystallographic symmetry of the sample.