I'm not convinced that the OP has two complete sets of collets and closers.
The upper set don't look like the Burnerd multisize collets I have. What I assume are the metal fingers are tapered, so they couldn't slide in slots like conventional multisize collets. They look more like rubberflex collets. Same result as sliding metal fingers but different construction. Rubberflex collets were often made by Jacobs, as were the ones I use in my auto reversing tapping heads. The lower set of collets are dead length collets.
The two types of collet work in different ways. Rubberflex collets work by contracting as they are pushed back towards the headstock. So the collet moves with respect to the headstock when it is closed. A corollary is that the work also moves relative to the headstock as the collet closes. Dead length collets are closed from behind, away from the headstock, towards a fixed stop at the front of the collet holder. A consequence is that the collet does not move relative to the headstock when being closed, nor does the work. Hence the name dead length. It's a significant advantage on repetition and capstan lathes.
It would be interesting to know if the two holders work in the same way, or different ways. The internal tapers should be different as well.
I think Martin is correct that some of the dead length collets are hexagon. I can vouch for their usefulness when making batches of nuts and bolts:
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Burnerd multisize collets will hold hexagons:
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But I'm not convinced the upper set of collets will, as I think the rubber between the blades would get in the way.
Andrew