Zanz,(the game designer) established the Material, Cardinal and Spiritual break downs, as far as I know.
I don't recall him ever establishing the three quartets as official. Bloodshadow used them in his writing in 2010 and over time it became the common interpretation of them, culiminating in the recent CIA rework using them.
There is his diagram
here [EDIT: accessible version
here] that establishes four pairs [
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] and a quartet [
], but it leaves the other two quartets up for discussion. Since the inner circle has the pairs across from each other (in that case, opposites, which are intuitive and the game confirms by pairing them on the selection screen), in theory the outer circle would also have the pairs across from each other (in that case, leading to the three quartets used today), but he's never officially said as much, to my knowledge.
Personally, I believe the trios, and CuCN just ninja'd me in saying this, should be orthogonal, or taking one from each quartet. There's a lot of different ways to interpret those and I don't really think any is more valid than all the others.
As for groups of six, the game splits the quanta interface into two halves [
] and [
]. Interestingly, if you connect the pairs on this interface, there's something of a pattern, and it doesn't seem random, so in theory it's possible that you could extrapolate a set of trios from there.
As for pulling a trio from these, it's worth noting that each third, from top to bottom, has three elements from a quartet, two on the left and one on the right. The other three on the right are misplaced. To me this is the most intuitive set of trios you can pull out, but there are probably more nuanced patterns in there as well.
Top Left:
(Steady)
Bottom Left:
(Destructive)
Right:
(Unbound)
Misplaced:
(Passive)
From Zanz's diagram linked above, you can also pull this set of trios (which I find slightly more difficult to name compared to the above, so I have not):
Western:
Northern:
Eastern:
Southern:
Additionally, crossing the quartets with the quanta-halves gives you six different sets of pairs besides the opposites, two within each quartet:
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. There's a lot of strong synergy in these pairings and I affectionately refer to them as the allies.
