user11432: Most computer chess games consider castling to be a king move. You click on the king and it shows an option to castle.

What do the FIDE's rules say about this though? Is castling a king move, a rook move, both, or neither?

user7415: 
As a purely separate thought on
the stupidity of the current rule
consider the following hypothetical case: 
White can castle, but both the f- and h-files are open.
White plays Rf1, suddenly realises he has missed a much
stronger move and quickly moves his king to g1 . As the rule
stands his opponent or, as happened in a recent Soviet event,
the arbiter can insist on a rook move. White calmly plays the
winning Rh1-h8. 

user7657: Castling is a combination move of both pieces.  The king should be moved first or both pieces simultaneously, since a rook move could stand alone and an opponent might make you stop your move at that point as being "legal", not letting you complete castling.

user12629: The FIDE rules say this about castling:

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


11432: <p>Most computer chess games consider castling to be a king move. You click on the king and it shows an option to castle.</p>

<p><a href="https://i.stack.imgur.com/V7jgt.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/V7jgt.png" alt="enter image description here"></a></p>

<p>What do the FIDE's rules say about this though? Is castling a king move, a rook move, both, or neither?</p>
 7415: <p><a href="https://i.stack.imgur.com/EDyPE.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/EDyPE.png" alt="enter image description here"></a></p>

<p>As a purely separate thought on
the stupidity of the current rule
consider the following hypothetical case: 
White can castle, but both the f- and h-files are open.
White plays Rf1, suddenly realises he has missed a much
stronger move and quickly moves his king to g1 . As the rule
stands his opponent or, as happened in a recent Soviet event,
the arbiter can insist on a rook move. White calmly plays the
winning Rh1-h8. </p>
 7657: <p>Castling is a combination move of both pieces.  The king should be moved first or both pieces simultaneously, since a rook move could stand alone and an opponent might make you stop your move at that point as being "legal", not letting you complete castling.</p>
 12629: <p>The <a href="http://www.fide.com/FIDE/handbook/LawsOfChess.pdf" rel="noreferrer">FIDE rules</a> say this about castling:</p>

<blockquote>
  <p>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</p>
</blockquote>
