Can we say anything about a graph $G$ if we know that its line graph $L(G)$ is Eulerian or Hamiltonian?

Question

Can we say anything about a graph $G$ if we know that its line graph $L(G)$ is Eulerian or Hamiltonian?

There's a lot of literature about the line graph of an Eulerian or Hamiltonian graph, but I can't figure out if you can say anything for sure going the other direction ($L(G) \to G$).