Most chess sets come with only one spare queen, and according to the video in the question Is holding the opponent's captured pieces legal or allowed?, one cannot use an upside-down rook because is effectively the same as promoting to a rook.
If two queens are already on the board, how can one promote a third one?
Under USCF rule 8F7:, it is acceptable to place a rook upside-down if a spare queen is not available.
If the desired piece is not available to replace a promoted pawn, the player may stop both clocks in order to locate that piece and place it on the board. A player who cannot quickly find such a piece may request the assistance of the director. It is common practice, however, to play using an upside-down rook for a second queen. In the absence of the player’s announcement to the contrary, an upside-down rook shall be considered a queen.
Under FIDE rules, you can't use an upside-down rook as a spare queen, but you can find the answer hidden in the section about the clocks:
6.11.2 A player may stop the chessclock only in order to seek the arbiter’s assistance, for example when promotion has taken place and the piece required is not available.
You pause the clock, call an arbiter and an additional queen will be provided.
