Precise instructions in a few paragraphs and you without tools, machinery or skills?
OK here goes...
Logs are first sent to a sawmill where they are cut into square timbers or a log-lathe to be turned round. Square timbers are then turned round on a lathe. (Without benefit of a lathe you'll need to "hew" them round as shipwrights have done more than 500 years ago.)
Round or tapered/round shafts are then laid-out with grid lines taken from the plan (ring and longitudinal lines). The intersection of these lines establish points along the spirals. Small nails are driven into these points and a chalk-line is wrapped round the shaft and snapped as you go (either that or a bendable straight-edge).
With all the spiral lines on the shaft, a hand or skill saw is used to make the initial cuts. These are made to depth along the spiral lines. (This is done on the lathe or between centers while the shaft is rotated.) Further cuts can be made to leave a spiral "V" notch. A hand saw can be fixed with a depth gauge or an antique stair saw could be used.
Now you're ready to carve, rasp and sand the loops of the rope. This is also done on the lathe. The shaft is rotated as you work but must also be solidly fixed as you carve. This is time consuming but not difficult. All the carving is done "downhill" of the grain. You'll need to work from opposite sides of the lathe, or left/right hand positions.
That's it.
I don't know what you're seeing in New Mexico but twist turnings are classified by style: rope, barley, Jacobean, open, etc. These all require different layouts but the execution of the work is the same.
Today twist turning are done on special CNC lathes like the Hapo 7000. Home-built lathes can be set-up like the old (Sears) router-crafter toy.