You could do something similar in "normal" functional programming languages, but you usually won't get the nice syntax.

Basically, you can represent each invertible function by a pair of functions (pseudo-Haskell, because I don't use it very often and can't test on mobile):

  data Invertible a b = Invertible { forward :: a, backward :: b }

  apply :: Invertible (a -> b) (b -> a) -> a
  apply f a = forward f a

  invert :: Invertible (Invertible a b -> Invertible b a) (Invertible b a -> Invertible a b)
  invert = Invertible { forward = invert', backward = invert'}
      where
      invert' :: Invertible a b -> Invertible b a
      invert f = Invertible { forward = backward f, backward = forward f}

Then you can for example invert invert itself by

  apply invert invert == invert

But because there is no built-in language support, you'll end up doing lots of parallel construction of the standard library before you can even think of using it productively. Apparently someone already did that for Haskell (https://hackage.haskell.org/package/invertible), but it likely doesn't cover everything.