  A graph is $(H_1,H_2)$-free for a pair of graphs $H_1,H_2$ if it contains no
induced subgraph isomorphic to $H_1$ or $H_2$. In 2001, Kr\'al',
Kratochv\'{\i}l, Tuza, and Woeginger initiated a study into the complexity of
Colouring for $(H_1,H_2)$-free graphs. Since then, others have tried to
complete their study, but many cases remain open. We focus on those
$(H_1,H_2)$-free graphs where $H_2$ is $\overline{H_1}$, the complement of
$H_1$. As these classes are closed under complementation, the computational
complexities of Colouring and Clique Cover coincide. By combining new and known
results, we are able to classify the complexity of Colouring and Clique Cover
for $(H,\overline{H})$-free graphs for all cases except when $H=sP_1+ P_3$ for
$s\geq 3$ or $H=sP_1+P_4$ for $s\geq 2$. We also classify the complexity of
Colouring on graph classes characterized by forbidding a finite number of
self-complementary induced subgraphs, and we initiate a study of $k$-Colouring
for $(P_r,\overline{P_r})$-free graphs.
