  Magneto-electric multipoles, which are odd under both space-inversion $\cal
I$ and time-reversal $\cal T$ symmetries, are fundamental in understanding and
characterizing magneto-electric materials. However, the detection of these
magneto-electric multipoles is often not straightforward as they remain
"hidden" in conventional experiments in part since many magneto-electrics
exhibit combined $\cal IT$ symmetry. In the present work, we show that the
anti-symmetric Compton profile is a unique signature for all the
magneto-electric multipoles, since the asymmetric magnetization density of the
magneto-electric multipoles couples to space via spin-orbit coupling, resulting
in an anti-symmetric Compton profile. We develop the key physics of the
anti-symmetric Compton scattering using symmetry analysis and demonstrate it
using explicit first-principles calculations for two well-known representative
materials with magneto-electric multipoles, insulating LiNiPO$_4$ and metallic
Mn$_2$Au. Our work emphasizes the crucial roles of the orientation of the spin
moments, the spin-orbit coupling, and the band structure in generating the
anti-symmetric Compton profile in magneto-electric materials.
