The Hardinge 24 (or other custom plates) indexer is on the left in this picture with an expansion collet. It is horizontal or vertical. Sometime called a "Tree" fixture in this area due to the casting source, they are available with a flat nose as seen here, (no external spindle extension), or with the Hardinge threaded nose or the Hardinge taper nose to take chucks and pot chuck closers for over size collets.
There's an egg case and an older square case Hardinge DH's in the picture, too. The TS is for the Tree fixture, it is too tall center height for the DH's.
The correct TS for Hardinge DH's is open straight through the casting, and has a 3-5/8" center height.
The Hardinge Dividing head on ebay posted above has a lot of issues.
It is missing the collet draw tube and the "wrench" - a sleeve with a knob wheel that friction fits in the drawtube to operate it.
The "knob wrench and the draw tube are shown assembled top left, and on the bottom right partially assembled with the additional strap wrench used to operate them. It will often cost more than a chuck, to acquire these parts separately, when they can be found at all.
The Ebay Hardinge DH is also missing the TS, another not easy to find accessory.
The clamp screw that is buggered up is not a simple deal. I have made them out of S7. Besides being an "interesting" job; dis-assembling, re-assembling and timing the parts is not trivial.
One thing I did on the new one was roto-broach the allen socket deeper and the next size larger. I "think" I made the journal a little larger too, and re-bored the top nut to match.
The eBay unit casting was broken and repaired at one time, down where the prongs for the plate clamp bears. Looks like both sides were broken given the size of the repair parts. That said, the repair looks like it was probably well done. Look at the last photo, and the let-in piece at the bottom left. There's also some "non-factory" excavation around the hold-down screw holes. Despite ad claim to the contrary, that DH has been beat up a bit; possibly dropped. Though there are several ways to break the clamp ring retainer casting.
Finally it appears to only have one 3 row division plate. Hardinge DH's are 4:1. So you need a lot of plates to get a full complement of dividing options. Or some of the older plates with more rows.
The chuck only has one set of jaws.
caveat emptor.
smt