One of Hilbert's many deceptively simple observations involved generators of hyperboloids of one sheet and hyperbolic paraboloids:
Three skew lines always define a one-sheeted hyperboloid, except in the case where they are all parallel to a single plane but not to each other. In this case, they determine a hyperbolic paraboloid (Hilbert and Cohn-Vossen 1999, p. 15).
This fact suggest the possibility that interesting transforms of these two surfaces into one another might be statable in terms of operations on triples of skew-lines.
Are there such transforms?
If not, is there some particular a priori reason why they can't exist?
I'm not asking this question idly, but rather as a follow-on to this question: