Up to now, all triple-differential cross-section measurements were performed with unpolarized electrons and atoms and, consequently, averaged over direct and exchange scattering. In our crossed-beam experiment, spin-polarized lithium atoms were ionized by spin-polarized electrons of 54.4 eV energy. In coplanar, asymmetric coincidence detection of scattered and ejected electrons, combined with electron-energy analysis, we measured a spin asymmetry which gives new information about the ionization process. Results are compared with recent calculations based on the distorted-wave Born approximation.