  Clique-width is an important graph parameter due to its algorithmic and
structural properties. A graph class is hereditary if it can be characterized
by a (not necessarily finite) set ${\cal H}$ of forbidden induced subgraphs. We
initiate a systematic study into the boundedness of clique-width of hereditary
graph classes closed under complementation. First, we extend the known
classification for the $|{\cal H}|=1$ case by classifying the boundedness of
clique-width for every set ${\cal H}$ of self-complementary graphs. We then
completely settle the $|{\cal H}|=2$ case. In particular, we determine one new
class of $(H,\overline{H})$-free graphs of bounded clique-width (as a side
effect, this leaves only six classes of $(H_1,H_2)$-free graphs, for which it
is not known whether their clique-width is bounded). Once we have obtained the
classification of the $|{\cal H}|=2$ case, we research the effect of forbidding
self-complementary graphs on the boundedness of clique-width. Surprisingly, we
show that for a set ${\cal F}$ of self-complementary graphs on at least five
vertices, the classification of the boundedness of clique-width for
$(\{H,\overline{H}\}\cup {\cal F})$-free graphs coincides with the one for the
$|{\cal H}|=2$ case if and only if ${\cal F}$ does not include the bull (the
only non-empty self-complementary graphs on fewer than five vertices are $P_1$
and $P_4$, and $P_4$-free graphs have clique-width at most $2$). Finally, we
discuss the consequences of our results for the Colouring problem.
