Since writing that post, I've had the opportunity to play a few such matches and now have some understanding of the techniques involved. You might assume that playing a two-game match with a random chess960 position is no different than playing a similar match with the traditional start position, but you would be wrong. In the traditional position, you don't care if your opponent opens 1.d4 or 1.e4 as White, or answers your 1.c4 with 1...c5, 1...e5, or any of the other choices. You've prepared for all eventualities and the initial moves are not about seeing which player has the better understanding of chess. That comes later in the game.
In chess960, on the other hand, already in the first moves you're comparing your initial plan as White with your opponent's plan in the other game. It's a contest to discover who has seen more possibilities in the given position.
The match against my strongest opponent started from SP077 NNRBKQBR. A little analysis convinced me that an early f2-f4 (...f5 for Black) is necessary. On top of letting the Bishop out, it gives the Queen a developing square, thereby preparing to castle O-O. The f4 idea was used by both players in both games. An early position in the game where I had Black is shown below, after the moves 1.f4 f5 2.Nc3 Nc6 3.Nb3 Nb6 4.e3 Qf6 5.Bf3. Note that White's last move prepares O-O-O.
What to play next? I looked at a number of ideas. The move 5...e5, developing the dark squared Bishop and setting up tension in the center, suggests itself. I rejected this because after 6.fxe5 Nxe5 7.Bxb7 Rb8 8.Ba6, it involves a Pawn sacrifice. Black gets good counterplay, but I wasn't convinced that it was enough for the Pawn.
More straightforward is 5...Bf7, but I didn't want to commit the Bishop to a passive square while more active possibilities were still available. A similar idea is 5...e6. This might have been the best choice, but I decided against it because of 6.Bxc6 dxc6 7.e4, when White's pieces look more active than Black's.
That last line gave me another idea. If White can give up the Bishop pair so soon, why can't Black? That's how I hit on the idea of 5...Bxb3. After the natural response 6.axb3, White has the open a-file, but it doesn't bring any immediate benefit. In return, Black is confident that White won't castle O-O-O, when the a-file is useless.
After I played 5...Bxb3, my opponent must have reasoned the same way, because he answered 6.cxb3. After a further 6...e6 7.O-O-O Be7, I was happy with my game, and eventually got a small positional plus. I still think 6.axb3 was better, but that would have been a different game.
In the traditional start position, you never have the opportunity to exchange a Bishop for a Knight on its first developing move (except after, e.g. Na3/Nh3, which is decidedly unusual). It's hard to overcome that sort of traditional thinking and look at a chess960 position with fresh eyes, but that's exactly what some positions require.