This is my proposed solution to the changes in update 2.4 which result in 2v2 games with the same team getting first move all the time; it's not perfect though and I'd appreciate any feedback.

1. Write out the initial move order. (I'll use an example of 7 to check it works as I go ABCDEFG)

2. Completely reverse the order, this is the move order for the next turn. (GFEDCBA)

3a. Swap the left most pair of letters (FGEDCBA) `Call this operation a 'pair swap'
3b. Swap the next left most pair of letters (FGDECBA)
3c. Continue swapping the next left pair until (n-1/2) pair swaps have been made, this is the move order for the next turn. (So FGDEBCA)

4. Return to step 2 and repeat the process

The example in full:
1. ABCDEFG
2. GFEDCBA
3. FGDEBCA
2. ACBEDGF
3. CAEBGDF
2. FDGBEAC
3. DFBGAEC
2. CEAGBFD
3. ECGAFBD
2. DBFAGCE
3. BDAFCGE
2. EGCFADB
3. GEFCDAB
2. BADCFEG
3. ABCDEFG - At this point, the move order has returned to its inital setting and so the entire cycle can start again.

Each player moves first twice and last twice (not including the return to original).
The biggest issue I can see is that the same player is last two turns in a row, but this could be changed so that either the same player is first two turns in a row (by starting the pair swaps from the right not the left) or maybe alternating right and left, but at this point I don't know if that'd affect the cycle (I'll perhaps do further examples to find out)

[Edited to display the full example from start to finish for the sake of clarity]

Edited 3/21/2014 21:24:17

"Each player moves first twice and last twice"
F and G only move one time first, and A never moves last.

Apologies, the bit after 'continue the example' does not include the first three moves, they were included in brackets further up at the ends of steps 1, 2 and 3c. These are ABCDEFG, GFEDCBA and FGDEBCA (followed by ACBEDGF and the rest). So, in fact F and G do move first twice, and the two times A is last are the two you didn't see.
Sorry, I should've made it more clear and written out the full sequence.