For spin-polarized electrons in the energy range of 8-21 eV incident off-normally on a W(110) surface, we have measured the energy distribution of secondary electrons using a time-of-flight technique and a position-sensitive detector. Selecting the elastically scattered electrons in the specular direction, we obtained the spin asymmetry of the (00) low-energy electron diffraction beam as a function of the primary electron energy and incidence angle. Calculations on the basis of a relativistic multiple scattering formalism, with potential input derived from the self-consistently calculated ground state electronic structure, yielded (00) beam spectra in rather good agreement with their experimental counterparts. In particular, we found a prominent asymmetry feature of about 60% slightly below the emergence threshold energy for two nonspecular beams. Its physical origin is a region of strong spin-orbit coupling between even and odd bulk states, but its size, sign, and energy depend sensitively on the surface potential barrier, which identifies it as a surface resonance. Experimentally, the surface sensitivity of the large asymmetry is revealed by its sign reversal after oxygen exposure.