Recursion Schemes - Haskell

Is there a recursion scheme that builds up a new structure while folding?
Basically I want something like (f a -> (a, f b)) -> Fix f -> (a, Fix f)

ie a bottom up traversal like cata, but returning a new tree using the info from the children

That signature should be (Fix f -> f (a, Fix g) -> Fix g) -> Fix f -> (a, Fix g)

And now having looked at it more I see that this is just para :face_palm:

And now that I am working with it, I notice that I do not even need the old structure, so cata does the job too