Mate with both hands: Result? Constructed case: In a tournament (LoC Tournament Rules apply, no rapid, no blitz) White plays d7-d8Q by placing the new Queen on d8 with the right hand and removing the pawn d7 with the left. White did not press the clock but claimed a checkmate.
Relevant rules:

4.1 Each move must be played with one hand only.
5.1.1 The game is won by the player who has checkmated his opponent’s king. This
immediately ends the game, provided that the move producing the checkmate position
was in accordance with Article 3 and Articles 4.2 – 4.7.
6.2.1 During the game each player, having made his move on the chessboard, shall stop his
own clock and start his opponent’s clock (that is to say, he shall press his clock). This
“completes” the move. A move is also completed if:

6.2.1.1 the move ends the game (see Articles 5.1.1, 5.2.1, 5.2.2, 9.6.1 and 9.6.2), or
6.2.1.2 the player has made his next move, when his previous move was not completed.


7.5.4 If a player uses two hands to make a single move (for example in case of castling,
capturing or promotion) and pressed the clock, it shall be considered and penalized as if
an illegal move.

I discussed the case with some IA and FA. In our opinion, the checkmate and the result 1-0 stands. White did not press the clock, so 7.5.4 does not apply. To complete the move, it is not necessary to press the clock because the move ends the game (6.2.1.1). 5.1.1 allows  a violation of 4.1 by not mentioning it.
We further think that that is not what was intended. We think that what was intended is

7.5.4 If a player uses two hands to make a single move (for example in case of castling,
capturing or promotion) and completes the move, it shall be considered and penalized as if
an illegal move.

Are we wrong with our decision? Is there a case illustrating that the formulation of 7.5.4 is according to it’s intention? Any other opinions?
My opinion (!) is that the rules are clear and correct.
Checkmate is not allowed when the move involves moving a piece in a way that the piece may not move (e.g. moving a bishop like a knight) and not allowed when touch-move rules are broken.
However checkmate IS allowed when a procedural rule which is intended to stop clock cheating is broken. This is because pressing the clock in the case of checkmate is not required. The two-handed rule is intended to tidy up the requirement that the clock is pressed with the same hand that makes the move. In cases where pressing the clock is irrelevant then so too is the use of two hands. Hence why 4.1 is deliberately and carefully excluded from 5.1.1
The rules commission obviously thought of this and, collectively, they are cleverer than both you and me :-)
Note, too, that the same logic applies to stalemate and &quot;dead positions&quot;. This careful thought by the rules commission is reflected in the relevant articles:

5.2.1  The game is drawn when the player to move has no legal move and his king is not in check. The game is said to end in ‘stalemate’. This immediately ends the game, provided that the move producing the stalemate position was in accordance with Article 3 and Articles 4.2 – 4.7.


5.2.2 The game is drawn when a position has arisen in which neither player can checkmate the opponent’s king with any series of legal moves. The game is said to end in a ‘dead position’. This immediately ends the game, provided that the move producing the position was in accordance with Article 3 and Articles 4.2 – 4.7.

There is one further area that this clarifies. Suppose that the player delivering checkmate with two hands has previously made an illegal move and does press the clock after delivering checkmate. They still win the game because the game ended with the checkmate, before the completion of the move with the clock press. Hence the second illegal move was made but not completed and only completed illegal moves are punished.