  We identify the triple-Q (3Q) state as magnetic ground state in Pd/Mn and
Rh/Mn bilayers on Re(0001) using spin-polarized scanning tunneling microscopy
and density functional theory. An atomistic model reveals that in general the
3Q state with tetrahedral magnetic order and zero net spin moment is coupled to
a hexagonal atomic lattice in a highly symmetric orientation via the
anisotropic symmetric exchange interaction, whereas other spin-orbit coupling
terms cancel due to symmetry. Our experiments are in agreement with the
predicted orientation of the 3Q state. A distortion from the ideal tetrahedral
angles leads to other orientations of the 3Q state which, however, results in a
reduced topological orbital magnetization compared to the ideal 3Q state.
