At a tournament last weekend, one of the players would move his piece with one hand, while removing a captured piece with the other. I did not interfere, as I know of no rule against this. The other player complained after the game, even though he had won. He insisted there was such a rule, but he could not find it either. In fact, the touch move rule discussed the situation in which a player simultaneously touches one of his own pieces and one of his opponents pieces.
See your 5th edition of the rules on page 62:
16c1. Using the clock. Each player must operate the clock with the same hand that moves the pieces.
Tim Just
5th Edition Editor
So then in such a situation that would seem to require that the player should use both hands to operate the clock. 
There’s nothing illegal about the capture. TJ’s rule quote refers to the clock. Especially in a blitz game, moving a piece with one hand and operating the clock with the other is a way to speed up stopping your clock and delaying time forfeit. It’s not allowed. I don’t see the analogy with capturing pieces, though.
Hmm… I should have been more clear. The rule I did quote was probably the one the complainer was thinking about (perhaps?). There is no specific rule regarding the “2 handed” capture.
Tim
Better not be…what if you have small hands and pieces are large?
I spend signifcant time with my students teaching them to play chess with one hand.
Based on an analogy with rule requiring one hand to be used for both piece movement and clock, I would tend support any claim regarding an opponent using one hand to move and the other to capture. Probable enforcement progression: Warning, 2 minutes added to opponents time, 5 minutes subtraction from offending player time and then game forfeit
Chess is essentialy a one handed game and the continual use of two hands is anoying to most opponents. Common sense tells me so and I don’t need a 1628 page rule book to spell it out for me.
Similiarly common sense, tells me than a using two hands for a piece promotion is unlikey to be a continual problem for the opponent and I would not disrupt a game for that claim.
I find this whole thread rather curious, as I’ve never heard this issue raised. In my 35 years of competitive chess, I have almost always picked up the captured piece with one hand, moved my piece to that square with the other hand, and then pressed the clock with whichever hand is closest. I don’t do it for speed as much as for coordination, and my two capturing motions are consecutive, not concurrent.
I’ve never paid attention to how my opponents do it, but I don’t think very many use the same hand for the capture. As long as they press the clock with a hand involved in the move, I’m OK with it.
Honestly, how many people have experienced complaints on this. Is this primarily a scholastic thing?
Really? I don’t find that annoying. What’s the big deal here?
Some players like to knock over the piece they are taking.
Some players remove it first and place it on the side, then put their piece on that square. I had an opponent once take my piece off, then mid-move re-considered which piece of his to move there, since he had two choices. Now THAT was a bit weird, us sitting there with this empty square waiting for a piece to occupy it. Took him a minute to finally decide. He really should have put my piece back, acknowledged that he has to take it, then do his thinking.
These are minor items in my opinion. The annoying stuff is continually adjusting pieces that don’t need it, or purposefully placing pieces at the edge of squares. I was at a US Open where a player sat down at a set up board with all four knights pointed at him and promptly turned all four knights toward his opponent. Immediately an argument broke out over that! Game hadn’t even started! I think Harry Sabine had to deal with that one. I remember reading in Chess Life some years ago about the idea of having your queen knight looking left, and the king knight looking right (for White), and a similar set up for Black. That way during the course of the game you can identify which knight that originally was. I liked that idea and adopted that style for myself.
This was not a scholastic tournament, and both players were adults.
Some set of rules or other (maybe WBCA) insists that, even when capturing, only one hand may be used. Curiously, castling is mentioned as the only exception, when two hands may be used.
If two hands are to be permitted when capturing, perhaps the rule should be that the clock must be operated with the same hand that moved the capturing piece.
Bill Smythe
Two handed chess players and former US champions.
youtube.com/watch?v=Bzrap8Vtyq8
Two handed captures are legal. It’s also never bothered me if a player has done a two handed capture. I’ve done them, particularly if I’ve been in time trouble. I’ve never thought about which hand I use to punch the clock if I’ve done a two handed capture. Then again I tend to switch hands that I use to move capture. I’m a righty, but I play left handed when the clock is on my left. Also it’s a matter of which hand is closer to whatever piece I’m moving.
I too have encountered a player who would take the captured piece off first and then pause before putting a piece on the abandoned square. It was rather disconcerting at first. Since he was a newbie I gently pointed out that the correct form to move the capturing piece first, then remove the captured piece.
I wonder if this is a regional thing? Many players around here remove the captured piece first, and then place the capturing piece on the square.
I’m not so sure it’s regional. I think it’s all in the timing. If someone grabs the captured piece first and then immediately replaces it with the capturing piece one may not even notice the order. But when they take the piece off, leaving a vacant square, and don’t place their piece on the square almost immediately it seems a little strange.
Gawd, now I’m going be watching how my opponents capture over the next few days to see if most players remove the captured piece before or after. 
I teach my students to remove the captured piece first. The reason is a tale of a Korchnoi game. (I’m pretty sure it was Korchnoi, anyway.) All he had to do was exchange rooks, and he would have a theoretically won endgame. He picked up the wrong Bishop to recapture, waved it in the air helplessly for a moment, then just walked out. Had he removed the rook first, picking up the wrong Bishop would not have been an issue.
While you’re at it, how about watching to see how players promote? I’d be willing to bet that a significant percentage (maybe even a majority) promote by first placing the queen on the eighth, then removing the pawn from the seventh. This is much less awkward than the alternative, pushing the pawn to the eighth, then removing it (perhaps in one continuous swell foop), then placing the queen on the eighth.
Bill Smythe
I pick up the queen and place her on the eighth, then remove the pawn from the seventh.
Geurt Gijssen, on ChessCafe.com, wrote about this perhaps a year ago. I understood Geurt to say it is fine to place the captured piece off the board onto the table, before then promptly moving the capturing piece to its new square.
I have never heard anyone say “correct form” is to move the capturing piece first, and I would disagree with such a claim (unless someone can cite a specific rule in the rule book).
I feel it is bad form to use more than one hand (per turn) to touch pieces and clock. It would not take much creativity to show knotty problem cases arising from such variability.
But I am not citing anything specific from the rule book. Nor do I care, unless perhaps a clock is down to its final seconds.
It is just that soooo much of the chess rule book is really about the clocks.
All things ‘clock’ should have their own section of the rule book, to remove the clutter from every other part of the rule book.