To play the Left-Right gift passing game, start by having everyone stand in a circle with the wrapped gifts that they brought. Someone then reads a story or poem filled with the words "left" and "right." Every time "left" is said in the story, each person passes his gift to the left; players do the opposite when "right" is said. At the end of the story, the gift that each person is holding is the one he opens and keeps.
Another gift exchange game to play is called Yankee Swap, in which people "steal" gifts from others. To play, each person brings a wrapped gift and places it in the middle of the room or area where the game is being played. Each person picks a number from a hat to determine the order of the gift picking.
The first person chooses a gift and opens it, letting everyone see what it is. The second player then chooses to steal the opened gift or open a new gift. This continues until everyone has had a turn to steal or choose a gift. When a person's gift is stolen, he can steal another gift or choose a new wrapped gift instead.
Another option is to play a Christmas trivia game. If someone answers a trivia question right, he gets to choose the next gift in the pile to open.