I've been thinking about your video, specifically about twisting direction: if you twist the string clockwise and then untwist it counterclockwise from the opposite point of view, are you not, from the original point of view, twisting clockwise the whole way through? The end result appears to work out, though, which makes this all the more confusing.
I'll try when I can with my Technic 3-ply gearbox that I haven't tested yet, because I'm curious to know what happens if you twist clockwise from the opposite point of view (which would make it counterclockwise from the original point of view).
Let me know if you try.