I think the pump-action screwdriver is a single-start thread in both directions. The lead is very coarse to allow the screw be driven by the nut – a lot less efficient than the conventional way round.
It is a very old idea, used in the Archimedean-type drill, probably pre-dating the 'Yankee' screwdriver, which I think was a trade-mark used by the American wood-work tool-makers, Stanley. I have such a drill, and found it useful for making pilot holes in hardboard for panel-pins when assembling a flat-kit book-case.
A development is the reversing-screw drive that oscillates something at constant speed (apart from a very short deceleration and acceleration at each end on the change-over). It consists of a pair of opposing square or Acme threads engaged by a cylindrical or pivoted rhomboidal follower that simply leaves one thread as it enters the other for the return trip.
Two applications I have known:
– Very short screws, only 2 or 3 turns, on a small lithographic printing-press, for oscillating the ink and water feed rollers against each to distribute the fluids evenly;
– Much longer ones operating the laying-fairlead on a winch built for handling what are called " towed arrays " – long hoses containing strings of hydrophones, used in naval defence and marine surveying. The hose has to be wound accurately to avoid damage, so the reversing-screw, parallel to the drum and geared from its drive, would have a lead (also pitch) equal to the hose diameter.
Although giving constant stroke length and speed, I have wondered if a reversing-screw could be one way to fit a power-drive to something like a manual shaper or planer, provided you have the equipment for milling hefty threads. Its advantage is that it would need only a simple motor and reduction-drive – no need for a reversing-switch, nor for stepper-motors and electronics.