I don't know the answer to this question, but the algebra of differential operators is almost commutative. So by looking at the principal symbols (with respect to the standard filtration by the degree of $D=\partial_x$), you can conclude that $\sigma(L_1)=\sigma(L_2)$. You can push it further a bit by considering other filtrations if, say, $L_i$ are polynomial coefficient operators, and by looking at the quasi-classical approximation given by the Poisson bracket.