I did some digging and found the universal operations mentioned in the last couple of posts: unzip, unbizip and counzip were referenced as abide, abide and coabide -- actually, I was looking for something else, and this fell into my lap.
They were apparently named for a notion defined by Richard Bird back in:
R.S. Bird. Lecture notes on constructive functional programming. In M. Broy, editor, Constructive Methods in Computing Science. International Summer School directed by F.L. Bauer [et al.], Springer Verlag, 1989. NATO Advanced Science Institute Series (Series F: Computer and System Sciences Vol. 55).
The notion can be summed up by defining that two binary operations and abide if for all a, b, c, d:
There is a cute pictorial explanation of this idea in Maarten Fokkinga's remarkably readable Ph.D dissertation on p. 20.
The idea appears again on p.88 as part of the famous 'banana split' theorem, and then later on p90 the above names above are given along with the laws:
fmap f &&& fmap g = unfzip . fmap (f &&& g)
bimap f g &&& bimap h j = unbizip . bimap (f &&& h) (g &&& j)
fmap f ||| fmap g = fmap (f ||| g) . counfzip
That said the cases when the inverse operations exist do not appear to be mentioned in these sources.