Gibb-

If some were pushing and some were pulling on four bars of a screw jack, wouldn't they be getting in each other's way?

We used to have an apple cider press that worked on a similar principle but only required two people exerting effort on a bar that went through the axle; in that case one person pulled while the other pushed, hopefully in coordination, and they wouldn't get in each other's way. But they would switch off grips as the bar worked its way around.

I suppose the cotton screwing gangs would also have to switch grips and reposition themselves as the bars came round. Otherwise they would end up tripping or stepping over the arm that was pressing in the bale, which seems more awkward.

In the Bosun's Locker, edited by Stan Hugill, p. 202, there's a drawing he made of what he thought cotton-screwing looked like. There're only two people working this screw-press but they are definitely "heaving" rather than "hauling."

Hugill, p. 203, also quotes Nordhoff as mentioning several screwing "chants" such as "Old Stormy," "Bonnie Laddie Hieland Laddie," and one other which I'm not sure we've mentioned:

Hurrah, Bee-man do!
Oh, we work for a Yankee dollar,
Hurrah, see-man do!

Cheerily,
Charley Noble