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Only when we take the diagonal of all 3 directions do we get the symmetry we want between the circles, meeting each of them in the same way that both tapers and twists, similar to the way Dini's surface (https://mathcurve.com/surfaces.gb/dini/dini.shtml) meets a straight line.
Note that for visibility I am trimming the surface to show a finite region in my animations, but it actually spirals through all of space, with the only singularities being this symmetrical pair of linked circles. Also, like with the loxodrome, we can vary the angle to change how many turns it takes as it goes from one circle to the other, more turns producing a denser filling of space.

The first post in this thread links to a rotatable 3d model of the surface. It's quite hard to grasp the shape from just an image or even an animation, I encourage you to turn it around yourself and try following a path from one circle to the other.

Here's another version of the model with a slightly tighter wrapping angle:
https://skfb.ly/puJDz