But almost. It has a vertical axis, lots of horizontal lines, and it twists.
But it is part of something bigger, a triply periodic minimal surface. 32 copies of the above piece, replicated by rotations and reflections, look like this:
This surface sits in a rectangular box over a square. If you identify top and bottom edge of the original squarical helicoid, you get a doubly twisted annulus, which is intimately (confomally, that is) related to a hollow spiderweb:
When squeezing the height down, our non-helicoids become even more helicoidal. When pulling the height up, the helicoids disappear. What we have here is a deformation of the Diamond surface of Hermann Amandus Schwarz.
When he sees this, he will probably just nod.
What happens when we pull a little further? We see doubly periodic Scherk surfaces emerging, stacked on top of each other.