  In vortex-like spin arrangements, multiple spins can combine into emergent
multipole moments. Such multipole moments have broken space-inversion and
time-reversal symmetries, and can therefore exhibit linear magnetoelectric (ME)
activity. Three types of such multipole moments are known: toroidal, monopole,
and quadrupole moments. So far, however, the ME-activity of these multipole
moments has only been established experimentally for the toroidal moment. Here,
we propose a magnetic square cupola cluster, in which four corner-sharing
square-coordinated metal-ligand fragments form a noncoplanar buckled structure,
as a promising structural unit that carries an ME-active multipole moment. We
substantiate this idea by observing clear magnetodielectric signals associated
with an antiferroic ME-active magnetic quadrupole order in the real material
Ba(TiO)Cu$_4$(PO$_4$)$_4$. The present result serves as a useful guide for
exploring and designing new ME-active materials based on vortex-like spin
arrangements.
