  The antiferromagnetic compound Ba(TiO)Cu$_4$(PO$_4$)$_4$ contains square
cupola of corner-sharing CuO$_4$ plaquettes, which were proposed to form
effective quadrupolar order. To identify the magnetic structure, we have
performed spherical neutron polarimetry measurements. Based on symmetry
analysis and careful measurements we conclude that the orientation of the
Cu$^{2+}$ spins form a non-collinear in-out structure with spins approximately
perpendicular to the CuO$_4$ motif. Strong Dzyaloshinskii-Moriya interaction
naturally lends itself to explain this phenomenon. The identification of the
ground state magnetic structure should serve well for future theoretical and
experimental studies into this and closely related compounds.
