user4418: As a follow on from the question Is it legal to castle with a "ghost" rook: When is it legal (according to the FIDE Laws of Chess) to castle moving the rook first?

user4418: The 2019 Champions Showdown: Chess 9LX (aka Chess 960) currently taking place in St. Louis should have been a clue.
Here is the relevant extract from the FIDE Laws of Chess:

Guidelines II. Chess960 Rules
II.3 Chess960 castling 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. However, a few
  interpretations of regular chess rules are needed for castling,
  because the regular rules presume initial locations of the rook and
  king that are often not applicable in Chess960.
II.3.2 How to castle. In Chess960, depending on the pre-castling
  position of the castling king and rook, the castling manoeuvre is
  performed by one of these four methods:
II.3.2.1 double-move castling: by making a move with the king and a
  move with the rook, or
II.3.2.2 transposition castling: by transposing the position of the
  king and the rook, or
II.3.2.3 king-move-only castling: by making only a move with the king,
  or
II.3.2.4 rook-move-only castling: by making only a move with the
  rook.

Here is one of the starting positions from yesterday's games which gives the possibility of rook-only castling on the queenside:
[fen "nrkrqbbn/pppppppp/8/8/8/8/PPPPPPPP/NRKRQBBN w KQkq - 0 1"]

The kings are already on the queenside castling squares of c1 and c8. All that is required is for the d file rooks to move and castling can be performed by hopping the b rook over the king.

user7297: Unless this is a trick question, I'd say never:

3.8.2 [] This is a move of the king and either rook of the same colour along the player’s first rank, counting as a single move of the king and executed as follows: the king is transferred from its original square two squares towards the rook on its original square, then that rook is transferred to the square the king has just crossed.

The word then implies the king moves first, and the rook second. Also,

4.4 If a player having the move:
4.4.1 touches his king and a rook he must castle on that side if it is legal to do so
4.4.2 deliberately touches a rook and then his king he is not allowed to castle on that side on that move and the situation shall be governed by Article 4.3.1

That would imply that touching both king and rook (but not moving them) commits the player to castling; then, he/she could then move the rook first and only then the king, since that would give the same position on the board as the only legal play at that moment. Still, if I would see that as an arbiter, I'd have a word with the player about this.

4418: <p>As a follow on from the question <a href="https://chess.stackexchange.com/q/25113"><em>Is it legal to castle with a "ghost" rook</em></a>: When is it legal (according to the <a href="https://www.fide.com/fide/handbook.html?id=208&amp;view=article" rel="noreferrer">FIDE Laws of Chess</a>) to castle moving the rook first?</p>
 4418: <p>The <a href="https://www.uschesschamps.com/2019-champions-showdown-chess-9LX/overview" rel="nofollow noreferrer">2019 Champions Showdown: Chess 9LX</a> (aka Chess 960) currently taking place in St. Louis should have been a clue.</p>

<p>Here is the relevant extract from the FIDE Laws of Chess:</p>

<blockquote>
  <p><strong>Guidelines II. Chess960 Rules</strong></p>
  
  <p><strong>II.3 Chess960 castling rules</strong></p>
  
  <p>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. However, a few
  interpretations of regular chess rules are needed for castling,
  because the regular rules presume initial locations of the rook and
  king that are often not applicable in Chess960.</p>
  
  <p>II.3.2 How to castle. In Chess960, depending on the pre-castling
  position of the castling king and rook, the castling manoeuvre is
  performed by one of these four methods:</p>
  
  <p>II.3.2.1 double-move castling: by making a move with the king and a
  move with the rook, or</p>
  
  <p>II.3.2.2 transposition castling: by transposing the position of the
  king and the rook, or</p>
  
  <p>II.3.2.3 king-move-only castling: by making only a move with the king,
  or</p>
  
  <p>II.3.2.4 <strong>rook-move-only castling: by making only a move with the
  rook</strong>.</p>
</blockquote>

<p>Here is one of the starting positions from yesterday's games which gives the possibility of rook-only castling on the queenside:</p>

<pre><code>[fen "nrkrqbbn/pppppppp/8/8/8/8/PPPPPPPP/NRKRQBBN w KQkq - 0 1"]
</code></pre>

<p>The kings are already on the queenside castling squares of c1 and c8. All that is required is for the d file rooks to move and castling can be performed by hopping the b rook over the king.</p>
 7297: <p>Unless this is a trick question, I'd say <strong>never</strong>:</p>
<blockquote>
<p>3.8.2 [] This is a move of the king and either rook of the same colour along the player’s first rank, counting as a single move of the king and executed as follows: the king is transferred from its original square two squares towards the rook on its original square, then that rook is transferred to the square the king has just crossed.</p>
</blockquote>
<p>The word <strong>then</strong> implies the king moves first, and the rook second. Also,</p>
<blockquote>
<p>4.4 If a player having the move:</p>
<p>4.4.1 touches his king and a rook he must castle on that side if it is legal to do so</p>
<p>4.4.2 deliberately touches a rook and then his king he is not allowed to castle on that side on that move and the situation shall be governed by Article 4.3.1</p>
</blockquote>
<p>That would imply that touching both king and rook (but not moving them) commits the player to castling; then, he/she could then move the rook first and only then the king, since that would give the same position on the board as the only legal play at that moment. Still, if I would see that as an arbiter, I'd have a word with the player about this.</p>
