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 $$ a=b=0\Longrightarrow\text{ {$a,b$}>0}, $$ where {$a,b$} is the Poisson bracket. 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 examples with more Poisson brackets.