The simultaneous ejection of two electrons from the (001) surface of W due to the collision of incident low-energy electrons with valence electrons has been studied experimentally and theoretically. Energy and momenta of the ejected electrons were measured simultaneously by a combination of coincidence and time-of-flight techniques. Calculations were performed in a relativistic distorted-wave Born approximation including exchange, in which the primary electron and the two emitted electrons are described by quasiparticle multiple scattering states. The valence electron is represented by linear combinations of Bloch waves matched at the surface. Screened Coulomb interaction matrix elements between these four states are evaluated. Experimental and calculated energy distributions from W(001) for very-low-energy primary electrons at normal and grazing incidence are in fairly good overall agreement. Although some features of one-dimensional bulk densities of states are roughly reflected, Coulomb matrix elements with low-energy-electron-diffraction-type states play a vital role. Further analysis reveals in detail the importance of elastic scattering of the primary electron and of the two ejected electrons. Some observed features can be attributed to occupied surface states.