The term "cross processing" comes from the days of film when professional photographers processed slide film in colour negative processing chemicals and vice-versa.
Perhaps this happened accidentally, originally, and the interesting colour balance was adopted as a way of enhancing an image.
Today you can emulate that effect in Photoshop and other digital image editing software.
The cross processing effect has to be used with subtlety as when overused it loses its impact and just looks plain wrong.
Giane has a page of cross processed cat photographs, which I am sure that you will enjoy.