Harnessing the spin of single atoms is at the heart of quantum information nanotechnology based on magnetic concepts. By attaching single Co atoms to monatomic Cu chains, we demonstrate the ability to control the spin orientation by the atomic environment. Due to spin–orbit coupling (SOC), the spin is tilted by ≈58° from the surface normal toward the chain as evidenced by inelastic tunneling spectroscopy. These findings are reproduced by density functional theory calculations and have implications for Co atoms on pristine Cu(111), which are believed to be Kondo systems. Our quantum Monte Carlo calculations suggest that SOC suppresses the Kondo effect of Co atoms at chains and on the flat surface. Our work impacts the fundamental understanding of low-energy excitations in nanostructures on surfaces and demonstrates the ability to manipulate atomic-scale magnetic moments, which can have tremendous implications for quantum devices.