Eulerian and Hamiltonian Graphs · Challenge problem

Split $6$ people into two groups of three, $L=\{1,2,3\}$ and $R=\{4,5,6\}$, and suppose every person in $L$ is friends with every person in $R$ (and no one is friends within their own group) — the complete bipartite graph $K_{3,3}$. Counting two seatings around a circular table as the same arrangement if one is a rotation or reflection of the other, how many circular seatings have every neighbor pair friends?