Symmetries in nanoscale structures can be decisive for their structural, electronic, and magnetic properties, particularly in systems with reduced dimensions. Here we show that the symmetries of a flat metal-organic molecule adsorbed on a transition metal dichalcogenide, a 2-dimensional layered material, have a dramatic effect on the total spin and the intramolecular spin-spin interactions. Using a scanning probe microscope, we find two different molecular spin states by modifying the symmetry of the molecules via the twist angle to the substrate. Additionally, we observe significant non-collinear Dzyaloshinskii-Moriya interaction between two electron spins on the molecule induced by the spin-orbit coupling of the van der Waals coupled layered material with broken inversion symmetry. Our work opens a path for modifying the spin by exploiting symmetries and for studying the nature of surface-induced non-collinear spin-spin interaction within a single molecule which might allow the realization of more complex topological spin structures. Symmetry is key for magnetism of molecules as well as other nanostructures. Here, the authors tune the magnetic moment of a metal-organic molecule deposited on NbSe2 via the adsorption symmetry, and observe a non-collinear intramolecular spin-spin interaction.