Mounted games and gymkhana are both versions of pattern racing. Mounted games is done with english tack and ponies. Gymkhana is a western sport. Both pony mounted games and western gymkhana are competitive sports in the US Pony Club.
Polo has 2 different types of competition- outdoor/grass, and arena. Arena polo also has school/college teams.
The modern game of polocrosse originated in Australia. It is similar to lacrosse on horseback and is played outside in an open field. Polocrosse is one of the competitive sports in the US Pony Club.