The banning of left-handed playing in a game of polo is for safety reasons in order to avoid the likelihood of a head-on collision between players. As a left-handed player and a right-handed player head for the ball, they would not pass each other as they do in right-hand only games.
Left-handed people are not banned from the game; they must play right-handed. There are even some professional left-handed polo players who play right-handed. The use of the left hand or the mallet was banned in the 1930s. The rule was relaxed after World War II when there was a lack of players, but the rules were re-introduced in 1974.