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