It's helpful to note that (r U2)5 U2 = [r: [U2, r] [r2 U2: r]]
That is,
r
U2 r U2 r'
r2 U2
(r)
U2 r2
r'
Clearly the first and last moves are outer conjugates to complete the centers, the second piece is a commutator, the parenthesized move is the quarter turn. The only not so obvious part (well not to me, but...) is the r2 U2 conjugates.
If you start out with a solved cube and do U2 r U2 r' r2 U2, you will see that when you do (r) next, the 1x(n-2) center block in face B in slice r replaces the 1x(n-2) center block in face D in slice r of the same color. In fact, that's why this 4-cycle even works at all, because with these conjugates, r2 U2, around the quarter turn, we displace only half of the center pieces. This is similar to my main method for forming 2-cycles, regardless if it's wide turn based (the Holy Grail) or single slice turn based (conventional algs) because we do a 3-cycle of three fourths of a slice and do a quarter turn to put back half of the center pieces.
The "Standard Dedge Flip" alg, or any other 2-cycle algorithm I have derived or my methods apply to can be derived this way too.
Since (r U2)5 U2 is 2-gen, the most popular dedge flip alg that can be modeled exactly like it is "Waffle's Alg" (I know that Sarah said Chester posted it before...anyway)
l U2 r' U2 r U2 l' U2
r U2
(r')
U2 r'
(Just conjugate all of that with r' U2 r2 to achieve a transformation of that alg).
The algorithm I posted previously is literally a 2-cycle of wings but with 3x3x3 distortion as well, not a 4-cycle (you probably know this, but it seems you are relating the two different cycles too closely). Proof: See the translation link to the 5x5x5.
If there is a 2-cycle algorithm of length 9 (or even less than 12) that can preserve the centers, much less everything but the LL, I certainly don't know how to approach it. I believe I have shown that I have a good understanding of (r U2)5 U2 to the extent that I linked my representation of it with my methods. Keep in mind also that (r U2)5 U2 is so brief because, by my outline, there were several move cancellations and the commutator was as brief as a commutator can be.
