More a comment than an answer.

I am not quite sure to understand the property you are looking for. After all the principal symbol of a pseudodifferential operator $P$ of order $m$ is a positively homogeneous function $p_m$ with degree $m$ on the pointed cotangent bundle and you may ask for some property of that principal symbol, indeed such as ellipticity or principal type.

Let me give you what I believe is a significant example, not included in your classification. Consider a pseudodifferential operator $P$ of order $m$
with a complex-valued principal symbol $p_m=a+ib$ such that
$$
a=b=0\Longrightarrow\text{ {$a,b$}>0}.
$$
Then for $R$ of order $m-1$,
$P+R$ is subelliptic with loss of $1/2$ derivative in the following sense
$$
(P+R)u\in H^s_{loc} \Longrightarrow u\in H^{s+m-\frac12}_{loc}. 
$$
There are more example with more Poisson brackets.