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.
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