The complex correlation of structure and magnetism in highly coercive monoatomic FePt surface alloys is studied using scanning tunneling microscopy, x-ray magnetic circular dichroism, and ab initio theory. Depending on the specific lateral atomic coordination of Fe either hard magnetic properties comparable to that of bulk FePt or complex noncollinear magnetism due to Dzyaloshinski-Moriya interactions are observed. Our calculations confirm the subtle dependence of the magnetic anisotropy and spin alignment on the local coordination and suggest that 3D stacking of Fe and Pt layers in bulk L1_{0} magnets is not essential to achieve high-anisotropy values.