The Scherk tower is made with Scherk 2nd surface.

Your model resembles a Scherk tower, but reference is made to the Scherk-Collins surface (which is non-minimal)

so after rereading your post, I see that the "saddle surface ring" Scherk-Collins surface:
"can be formed by twisting and warping a singly-periodic Scherk’s minimal surface"
so I'm doing some more reading, both art and math... :-)

https://pdfs.semanticscholar.org/8ac1/d46735688959ec7f1f58a92dd268d0e1388c.pdf

The Scherk1

- Brian

I did try out the mathmod and sculpture generator programs a bit.

I'm going to try stacking some Scherk1 saddles using c=0.99, u = 25, v = 25, 100 pts/curve, trimming the tops and bottoms,
copying above and below, rotate 90 degrees, blending, then network...