What Other Bidding Approaches Are There?
“legal” is a meaningless question without knowing where you are, and, in many regions, some methods are only legal at higher levels of play.

Some of the more common uncommon system types (systems may also fit into more than one category):


Canapé: opening the shorter of your two suits. (How they bid one-suited and balanced hands varies.) This was used a fair amount by the Italian Blue Team in their heyday, and Blue Team Club is an example.
Non-strong forcing openings: There’s a forcing opening that isn’t necessarily strong. Polish Club is a well-known example.
Forcing pass: pass promises strength (often opening values), and most or all initial bids therefore deny it. There’s usually one opening bid reserved for the truly awful hands, and the rest show some strength. These methods are banned in a lot of regions, and are (IIRC) fairly regulated even in international events.
Strong pass: Pass contains both the terrible hands and some stronger, but limited hands. Many opening bids are still limited to intermediate values, but the initial pass is itself passable, so the systems are less weird. (AFAIK, these methods are pretty uncommon.)
Relay methods: Don’t govern the opening bid as such, but in some auctions (generally strong ones) the players don’t mutually describe their hands. Instead, one player uses atrificial bids, often simply the next available bid, and their partner responds, providing succesively more information until their partner thinks they can place the contract.


I’m sure I’m forgetting some broad categories.

In addition, there are lots of individual systems whose designers decide that some particular attribute of the hand should be emphasized in some way, and build a system around that. So, you might have a system where 1C and 1D deny or promise a 4-card major; or a system where opening 1 major denies 5 cards, showing 4 or 6+; or a system that opens 4-card majors, but only on minimum opening hands.