  We describe a method to compute the Brauer-Manin obstruction for smooth cubic
surfaces over $\bbQ$ such that $\Br(S)/\Br(\bbQ)$ is of order two or four. This
covers the vast majority of the cases when this group is non-zero. Our approach
is to associate a Brauer class with every Galois invariant double-six. We show
that all order two Brauer classes may be obtained in this way. We also recover
Sir Peter Swinnerton-Dyer's result that $\Br(S)/\Br(\bbQ)$ may take only five
values.
