  Magnetic Compton scattering is an established tool for probing magnetism in
ferromagnetic or ferrimagnetic materials with a net spin polarization. Here we
show that, counterintuitively, {\it non-magnetic} systems can also have a
non-zero magnetic Compton profile, provided that space-inversion symmetry is
broken. The magnetic Compton profile is antisymmetric in momentum and, if the
inversion symmetry is broken by an electric-field switchable ferroelectric
distortion, can be reversed using an electric field. We show that the
underlying physics of the magnetic Compton profile and its electrical control
are conveniently described in terms of $k$-space magnetoelectric multipoles,
which are reciprocal to the real-space charge dipoles associated with the
broken inversion symmetry. Using the prototypical ferroelectric lead titanate,
PbTiO$_3$, as an example, we show that the ferroelectric polarization
introduces a spin asymmetry in momentum space that corresponds to a pure
$k$-space magnetoelectric toroidal moment. This in turn manifests in an
antisymmetric magnetic Compton profile which can be reversed using an electric
field. Our work suggests an experimental route to directly measuring and tuning
hidden $k$-space magnetoelectric multipoles via their magnetic Compton profile.
