Dressage, Show Jumping, and Cross Country are the styles of horseback riding but there may or may not have been some western riding in there...
Answer2:
Currently only Dressage, Show Jumping, and Eventing are Olympic sports. In the past the high jump (puissance) and long jump were included, but these events were dropped in favor of the top three events.
In the ancient Olympic Chariot Racing was involved, but this is also no longer included.
Western riding has never been an Olympic sport and attempts to have sports like reining included have been turned down multiple times by the Olympic committee.
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