  Given a finite simple graph $\Gamma$ on $n$ vertices its complementary prism
is the graph $\Gamma\bar{\Gamma}$ that is obtained from $\Gamma$ and its
complement $\bar{\Gamma}$ by adding a perfect matching, where each its edge
connects two copies of the same vertex in $\Gamma$ and $\bar{\Gamma}$. It
generalizes the Petersen graph, which is obtained if $\Gamma$ is the pentagon.
The automorphism group of $\Gamma\bar{\Gamma}$ is described for arbitrary graph
$\Gamma$. In particular, it is shown that the ratio between the cardinalities
of the automorphism groups of $\Gamma\bar{\Gamma}$ and $\Gamma$ can attain only
values $1$, $2$, $4$, and $12$. It is shown that the Cheeger number of
$\Gamma\bar{\Gamma}$ equals either 1 or $1-\frac{1}{n}$, and the two
corresponding classes of graphs are fully determined. It is proved that
$\Gamma\bar{\Gamma}$ is vertex-transitive if and only if $\Gamma$ is
vertex-transitive and self-complementary. In this case the complementary prism
is Hamiltonian-connected whenever $n>5$, and is not a Cayley graph whenever
$n>1$. The main results involve endomorphisms of graph $\Gamma\bar{\Gamma}$. It
is shown that $\Gamma\bar{\Gamma}$ is a core, i.e. all its endomorphisms are
automorphisms, whenever $\Gamma$ is strongly regular and self-complementary.
The same conclusion is obtained for many vertex-transitive self-complementary
graphs. In particular, it is shown that if there exists a vertex-transitive
self-complementary graph $\Gamma$ such that $\Gamma\bar{\Gamma}$ is not a core,
then $\Gamma$ is neither a core nor its core is a complete graph.
