Excellent question. The rules state that this is not possible. Castling can only be done once.
This is explicitly mentioned under Guidelines II. Chess960 Rules:
II.3.1 Chess960 allows each player to castle once per game, a move by potentially both the king and rook in a single move. [...]
Specifically, castling (even if the king doesn't move) should still be done by touching the king before moving the rook. You can do this several ways but the standard recommendation is to move the king off the board, move the rook to the new location, and move the king back to the new position (which can be the original c1 square).
II.126.96.36.199 When castling on a physical board with a human player, it is recommended that the king be moved outside the playing surface next to his final position, the rook then be moved from its starting position to its final position, and then the king be placed on his final square.
II.188.8.131.52 To avoid any misunderstanding, it may be useful to state "I am about to castle" before castling.