  Using combined theoretical and experimental approaches, we studied the
structural and electronic origin of the magnetic structure in hexagonal
LuFeO$_3$. Besides showing the strong exchange coupling that is consistent with
the high magnetic ordering temperature, the previously observed spin
reorientation transition is explained by the theoretically calculated magnetic
phase diagram. The structural origin of this spin reorientation that is
responsible for the appearance of spontaneous magnetization, is identified by
theory and verified by x-ray diffraction and absorption experiments.
