My opponent took a move back. I was about to take his Queen, but had to catch a plane. Next time I looked, he took back his move and made another.
The FAQ says you can't do that.
Can I take a move back/reinstate a timed out game if my opponent agrees?
Unfortunately, no. Once a move has been made it cannot be taken back. It is possible to start a new game from any position though. To do this you can recreate a position by using the 'Set Piece' facility when creating a new game.