I believe it's an early slimline pen mandrel, little like a mandrel saver.
If I remember the small piece used to have a bearing fitted in/on the small Diameter end to allow the shaft to rotate easily.
If I remember rightly Craft Supplies UK used to sell them when they were based at Millers Dale.
Both the ends are tapered to an MT2 and the shaft should be around 70mm long.
That was based on adding your Bushes and turn one tube at a time.
I believe that the bearings used to fail very quickly and was a poor design as the bearing was a ball race type that wasn't designed to have sidwards pressure..
That's my fourpenoth
I gather you must have purchased a bulk lot of stuff of e bay from someone who used to do wood turning.?