In the area of pseudo-differential operators, we know that for elliptic type or real principal type operators reductions are independent of lower order terms. For example, if $P$ is zeroth order real principal type operator with principal symbol $q(x,\xi),$ then for a lower order perturbation $R$ we can always find a zeroth order elliptic operator $E$ such that $E^{-1}(P+R)E=q(x,D).$

The reason for this independence of lower terms in elliptic type operator is because the principal term can be inverted and in real principal type operator is there exists a non-degenerate Hamiltonian flow.

Now my questions are:

Are there any other class of pseudo-differential operators which have this property? If not, is there any work in microlocal analysis which tells us that elliptic and real principal type operators form the largest such class?