A study of the elastic exciton-exciton Coulomb scattering in a semiconductor quantum well is presented, including the interexciton exchange of carriers and the spin degrees of freedom. The theoretical results show that electron-electron and hole-hole exchanges are the dominant mechanisms of interaction, while the classical direct term is negligible. The density-dependent homogeneous linewidth is calculated within the Born approximation and good agreement with the existing experimental data is obtained. Owing to the interexciton exchange of carriers, collisions lead to spin relaxation as actually observed in recent time-resolved photoluminescence experiments. [S0163-1829(98)06736-8].