For spin-polarized low-energy electrons impinging on non-magnetic crystalline surfaces, the collision with a valence electron and the ensuing emission of an electron pair are treated in a distorted-wave Born approximation formalism with exchange, in which the four relevant quasi-one-electron states are solutions of the Dirac equation. Numerical calculations for W(001), which were carried out in two coplanar geometries with normal and grazing incidence of a primary beam polarized normal to the reaction plane, show that the (e, 2e) cross section changes significantly upon reversal of the polarization. Originating mainly from spin-orbit coupling in the valence electron state, asymmetries of up to 30% occur in conjunction with sizable intensity. The calculated spectra respond sensitively to changes in the surface structure.