Is this relative major/minor inversion just Jim's thing? I've never heard of it before but it sounds a little interesting.
"Relative" goes with scales. Major and minor scales that have the same key signature are relative.
"Inversion" goes with intervals. For example, an M3 inverts to a m6. I have heard of inverting a chord, where a major chord inverts into a minor chord, but that was 20th century music.
It sounds like reharmonization, like Brian said.
If the original chord progression is Emi, CMaj, B7, that's a i, VI, V7 progression in e minor. Jim's relative major/minor inversion made it CMaj, Amin, G#min which is I, vi, #v min (or b vi more likely) -- not quite a progression anymore.
From what I've read, a Major chord inverts to a minor, right? and then you take the root of the chord that is a from the relative key if you think of that chord as the key? The dominant 7th is a little odd then -- It's not major or minor. What does that invert to?
Let me think...
I becomes vi, similar function... tonic.... Actually, you could probably just hear I+6
ii becomes VII, predominant and I'm not sure (V/iii or dominant to the tonic?)
iii becomes I, similar function.... tonic
IV becomes ii, similar function.... predominant
V becomes iii, not similar at all.... dominant to than pre-pre-dominant "floating" level
vi becomes IV, similar funtion... predominant
vii becomes

-- What would a diminished chord invert to? Augmented? If you mirror the intervals, a diminished chord inverts into a diminished chord.
So...
You're really just jumping up or down a third from the original root of the chord. There's a logic to it, so it probably has a consistent sound. I'm not sure it's always functional or going to work in every case. Must just be Jim's thing then.
It's work ok on a I, iii, IV, and vi chord because the function stays the same. Although for I and V, you could just be hearing the original chord with added tones. The dominant 7th chord was really treated like a Major chord the way Jim used it.
I'm guessing this a technique only Jim is using. If it's a full-fledged jazz idea, let me know because I want to think about it more.
I don't mean to insult you or anything Jim, but you might want to study reharmonization more then. You can always add 7ths to most chords. Maybe this technique is already doing that to some point. I just don't think it's going going to sound functional and smooth though. At least that's what I'm wondering. Maybe it's some inside jazz secret for doing quick reharmonization. If it's just your thing Jim, find out more about reharmonization because you'll probably find the results sound a little more polished.