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.

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?

Note: As stated in the question, my interest is in 'reductions of operators', i.e., transforming to simpler forms. I am not interested in regularity or solvability problems of pseudo-differential operators. I am aware of examples(not stated in the question) of operators for which the solvability issues are independent of lower order terms. For instance, http://annals.math.princeton.edu/wp-content/uploads/annals-v163-n2-p02.pdf