TITLE: Ordering of large cardinals by cardinality
QUESTION [16 upvotes]: Let Type A and Type B be two types of large cardinals from, say, Cantor's Attic (http://cantorsattic.info/Upper_attic)
Now assuming that ZFC + Type A + Type B is consistent (ie, both Type A and Type B cardinals can coexist), I define:
*Type A > Type B if smallest Type A cardinal has higher cardinality than smallest Type B
*Type A = Type B if smallest Type A and Type B have same cardinalities
*Type A $\perp$ Type B if the ordering in the sense above is undecidable 
So, for instance, Inaccesssible < Hyper Inaccessible < Mahlo
What is known about the ordering of large cardinals from (http://cantorsattic.info/Upper_attic) in this sense ?
I am particularly interested in the "large" large cardinals from measurable upwards.
For eg: How would one order measurable, extendible, huge and rank-into-rank ?
Motivation: I understand researchers are mostly focused on consistency strength, but I am interested in the intuitive notion of getting "much bigger infinities" from each successive large cardinal axiom.

REPLY [11 votes]: Let me add one extra example that might be interesting. 
Let $\pi_n^m$ and $\sigma_n^m$ denote respectively the least $\Pi_n^m$-indescribable and the least $\Sigma_n^m$-indescribable cardinal  (if they exist). Then:
Fact 1. If $V=L,$ then  $\sigma_n^m < \pi_n^m,$ for all $n, m \geq 1,$
Fact 2 (Hauser). If the existence of a $Σ^m_n$ indescribable above a $Π^m_n$ indescribable is consistent with $ZFC$, then the theory $ZFC+GCH+σ^m_n>π^m_n$ is consistent.