The electrons in 2D systems with broken inversion symmetry are spin-polarized due to spin-orbit coupling and provide perfect targets for observing exotic spin-related fundamental phenomena. We observe a Fermi surface with a novel spin texture in the 2D metallic system formed by indium double layers on Si(111) and find that the primary origin of the spin-polarized electronic states of this system is the orbital angular momentum and not the so-called Rashba effect. The present results deepen the understanding of the physics arising from spin-orbit coupling in atomic-layered materials with consequences for spintronic devices and the physics of the superconducting state.