If the gear is a worm and wheel, the smaller shaft will subject to an axial load anyway: that given by the action of the worm itself. Hence the thrust-bearing, and if there is only one the gear is intended to be driven only always the direction that loads the bearing.
Also, the smaller is the input shaft so it drives the pulley, with a fairly large reduction in speed.
Normally it is impossible to drive a worm from its wheel. It can be done with a very long worm lead giving a very high helix angle on the wheel, but at low efficiency. There are some applications for this, such as the speed-limiting “fly” in a musical-box movement, but little else.
Alternatively the drive is a helical-gear pair, but there does not seem room for that in this mechanism.
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Just looked again…..
Photo 1 gives just enough view of the larger gear to suggest it is not a worm-wheel.
Photo 2 gives a glimpse of the smaller gear. It is hard to tell but it does not look like a worm: the two diagonal lines suggest a high helix-angle more akin to a helical pinion; and teeth not a single “tooth”. (Worms can have 1, 2, 3 or 4 “teeth” (threads) but the angle on this one does not suggest even a single-start thread.)
So this may well be a helical-gear pair, which may drive in either direction. I think it will still give some axial load hence the thrust-bearing.
If so you should be able to rotate it by hand from either shaft. A worm drive would lock if you try to turn it from the pulley.
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I note too, that the pulleys appear to be for round-section belts.
The flat on the smaller shaft has a grub-screw bruise so perhaps from another pulley.
The small block on the top seems to give a small degree of rotary and/or adjustment, but not much.
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Is the shaft on its own, actually a “spare” for the assembly? Or do you have there the bulk of a tool-post accessory set designed for use both axially and radially to the lathe, assuming a plain top-slide not a QCTP with its two sets of slides? In that case the adjuster, which looks like an add-on, may have set the centre-height.