We identify the triple-$\mathbf{q}$ (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 would lead to other orientations of the 3Q state with a reduced topological orbital magnetization compared to the ideal 3Q state.

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• Magnetic anisotropy
• Magnetic interactions
• Magnetic order
• Magnetic texture
• Chiral magnets
• Interfaces
• Magnetic thin films
• Noncollinear magnets
• Density functional theory
• Scanning tunneling microscopy