Given a graph $G$ with vertex set $V$, let $HP(G)$ be the graph on $V$ where there's an edge from $u$ to $v$ if and only if there's a Hamiltonian path in $G$ from $u$ to $v$. (I believe this is called the Hamiltonian path operator, but all references I can find to it are from a computational perspective.)

Anyway, here's my question: for which graphs $G$ is $HP(G) \simeq G$? There are some obvious examples (complete graphs, their complements, *n*-cycles, and $K_{n,n}$s), and at least one less-obvious example ($K_{2,1,1}$, or a square with one diagonal). The latter is certainly more interesting, because the isomorphism between $HP(G)$ and $G$ doesn't just map each vertex to itself. Are there others like it, or any obvious ones I missed? Even better, can we completely classify these graphs?