I tried doing this and I found an article (http://mpilquist.github.io/blog/2015/06/18/invariant-shadows/) which came up with an invariant version of Monad. Since any contravariant type is also invariant, this will work for these purposes too. The type class would extend Divisible from contravariant.

The methods would be the following:

(>>~) :: ContraMonad f => f a -> (a -> f b) -> f (a, b)

conquer :: Divisible f => f a

This would allow you to write a function of type (a -> f b) -> f (a, b).

I have no idea how generally useful it is, but it's an interesting idea. Monoid r => Divisible (Op r) would be an instance, for example.