The cuprate superconductors exhibit remarkably-high superconducting transition temperatures (Tc) and remain the source of unsolved fundamental problems in quantum materials research after decades of intense study. In particular, the pseudogap phase, observed above Tc at low and intermediate doping, has been associated with a range of experimental signatures, but still no theoretical model that fully describes these results has been developed. One of these experimental signatures, intra-unit-cell (IUC) magnetism, is marked by the appearance of magnetic order within the unit cell in a manner such that there is a net cancellation of magnetic field. This order has the same periodicity as the crystal lattice despite the absence of a bulk magnetic moment as would be seen in a ferromagnet. The present polarized-neutron diffraction measurements were made by scientists from the Center for Quantum Materials (CQM) at University of Minnesota, Laboratoire Léon Brillouin (France), and Institut Laue Langevin (France). The work places significant new constraints on the orientation of the magnetic moment involved in the IUC magnetic order. These measurements were made on exquisite crystals of HgBa2CuO4+δ that were grown in the CQM. HgBa2CuO4+δ is a compound with a simple crystal structure and the highest Tc among cuprates with a single copper-oxygen plane per unit cell. The model nature of HgBa2CuO4+δ implies that the results are representative of the physics of the underlying copper-oxygen planes, relatively unmodified by the structural complications that are more prominent in other cuprates. This work places constraints on the orientation of the magnetic moment involved in the IUC magnetic order with unprecedented precision, ruling out several important theoretical models of the IUC magnetic order. By constraining existing models and providing detailed experimental data on which to base new theoretical work, this work helps illuminate the mysterious pseudogap phase of the cuprates.