The translation invariant Callahan-Hoffman-Meeks Surfaces can be deformed into screw motion invariant surfaces.
The straight lines survive this deformation, and allow for a relatively simple existence proof, which you can find in the paper by Callahan, Hoffman and Meeks.
In the standard case with dihedral symmetry 2 shown here, one vary the angle between the horizontal lines at different heights between 0 (the translation invariant case) and up to π/2, but not above. The picture above is close to the maximal twist and suggests the appearance of helicoids. This is further confirmed by a view from the top:
In the limit, we obtain a parking garage structure with four helicoidal columns at the vertices of a rhombus whose diagonals have the ratio 1+√2. This follows from the Traizet balance equation for helicoidal limits.
M. Callahan, D. Hoffman, W. Meeks: Embedded Minimal Surfaces With an Infinite Number of Ends, Inventiones Math. 96 (1989), 459-505.