  We present a combined theoretical and experimental investigation of the
anti-symmetric Compton profile in LiNiPO$_4$ as a possible probe for
magneto-electric toroidal moments. Understanding as well as detecting such
magneto-electric multipoles is an active area of research in condensed matter
physics. Our calculations, based on density functional theory, indicate an
anti-symmetric Compton profile in the direction of the $t_y$ toroidal moment in
momentum space, with the computed anti-symmetric profile around four orders of
magnitude smaller than the total profile. The difference signal that we measure
is consistent with the computed profile, but of the same order of magnitude as
the statistical errors and systematic uncertainties of the experiment. Our
results motivate further theoretical work to understand the factors that
influence the size of the anti-symmetric Compton profile, and to identify
materials exhibiting larger effects.
