In-N-Out is going all animal (style) on Smashburger for peddling a burger it considers triple the cheese, and double the theft ... so In-N-Out's suing.

In-N-Out just filed suit against its burger rival for allegedly jacking their double-double, triple-triple and quad-quad concepts, and tweaked it with their own version -- 3 cheese slices and 2 patties, or as they call it ... the Triple Double.

I-N-O acknowledges it didn't specifically trademark the name Triple Double -- although it claims its patrons are known to customize orders with similar terms. But in the suit, it says Smash's move is causing confusion in the marketplace and screwing with their burger biz.

We've reached out to Smash for comment ... so far, no word back.