There originally seems like there shouldn't be one, because it's impossible to, for any permutation of the arguments of (>>=), get the covariances/contravariances right on the arguments. However, based on the fact that there is a mapping:

a -> ((a -> r) -> r)

and what seems like an isomorphism:

(a -> r) -> (((a -> r) -> r) -> r)

(((a -> r) -> r) -> r) -> (a -> r)

then this looks like it might be some kind of monad-like structure:

class Contravariant f => Contramonad f where

contrareturn :: a -> f (f a)

contrabind :: (a -> f (f b)) -> f b -> f a

I have no idea what the laws should be for it, though. So is this worth pursuing, or have I just stumbled down a blind alley?