Just noticed that Hugill again refers to the screwing action as heaving (twice) in his SHANTIES AND SAILORS" SONGS (1969)...though that doesn't tell us much because I doubt if he ever saw the operation himself!

Perhaps the screw (how many handles? 2? 4, for each member of the "gang"?) were situated in such a way that the men stood on either side of it. Either it went 'round like a capstan (parallel to the ground) or like a vault/safe door handle (perpendicular to ground). Either way, if men stood on opposite sides, some would be heaving while others would be hauling.

Gibb