Let's Check the Rules

While I'm on the subject of castling, as in Posts with Label 'Castling', there was a relevant question a few months ago on the Chessexpress blog: Fischer Random Fun. Although it dealt with a specific position (SP748 RBKNNQBR),

Assume that White does castle Queenside [...] If White then moves the Bishop on g1, the Queen on f1, the Knight on e1, and the 'castled Rook' on d1 off the back rank, can White now castle Kingside, as the King and Rook have not yet moved?

the same question applies to any start position (SP) with King on c1 and the h-side Rook on any square except d1. It can also apply to any SP with King on g1 and the a-side Rook on any square except f1.

I'm no expert on the rules of chess960 (not to mention the rules of traditional chess), so my first reaction to this sort of question is always, 'Let's see what the rules say'. In 2009, the rules of chess960 were incorporated into the FIDE Handbook, under Laws of Chess :: Appendices (section: 'F. Chess960 Rules'). There we find

F.3 Chess960 Castling Rules: a. Chess960 allows each player to castle once per game, a move by potentially both the king and rook in a single move. [...]

That's clear enough. Since a player can only 'castle once per game', it's not possible to castle Queenside, then Kingside, or vice versa. As they say, 'When all else fails, read the instructions!'