You'll also want to check your local ordinances which can often be more restrictive than statewide laws. For example, though the language of the Minnesota law only seems to ban switchblades, the St. Paul weapons ordinance uses language that appears to ban AO knives as well (they mentioned "spring blade" knives in addition to switchblades). Additionally (again using St. Paul as an example), if the blade is too thin, it might be interpreted as a stiletto (a blade shape designed to stab), which is also banned. You also have blade length to worry about.
One thing I can tell you is that regardless of what the law says, you'd be in trouble if you were ever caught with it. Balisongs have a really bad public image and you'd have an incredibly hard time convincing any LEO that you weren't carrying it as a weapon, even if it was technically legal to carry. They just don't look like tools.
If all you want to do is keep it at home, then go nuts. Minnesota makes plenty of exceptions for the sake of collectors. However, if you were thinking about carrying it, you'd likely be in for a world of legal trouble if you were ever caught with it.