If it was just someone playing around with a lathe, would you leave the middle part squared off? I would think it would introduce a slight wobble to the workpiece and so you’d knock off the corners almost immediately.
And if you were making two of the same pieces on a lathe, would you mirror image them like that or would you have the two pieces facing the same direction?
I don’t know what it is but I think I’ve seen something like that in an antique store. If I’m right, it’s not unique.
I think the center section was left square so it wouldn’t roll away when it’s set down. That tells me that it’s a useful tool.
Kite-string holder (the acorns spin in your hands)
I think it’s either a piece of carving that fell off the top of a headboard or something – I have a bed very similar to this and pieces like the one you depicted keep falling off of it.
My second guess is some sort of fancy weaving shuttle. I see others were of the same mind.
Try the What is this thing? forum on Reddit; they figure out EVERYTHING! 
Looks ornamental to me. And acorns are only smooth on the part that is smooth. The caps of acorns are rough.
Seems like a tool.
The two smooth polished ends being the important parts. The center portion after the ridges being less important. The square center, so your tool does not always roll away.
Maybe you used it to align two pieces. Hold them so as you did something to them. Or it kept them apart but under control. A spreader.
Not very worn or scratched. So used with cloth? Or something soft. A temporary plug between two tubes? Just tossing out ideas to see if it triggers more ideas.
Definitely not a tool. You wouldn’t have that sort of finish on any sort of working surface, and the shape clearly has “looks like an acorn” as the primary design consideration, not secondary to some practical use. Plus, it looks like it’d be pretty fragile at the unfinished stems.
It’s something decorative. I’m wavering between a decorative piece that came off a piece of furniture, or just the result of someone idly fiddling around with a lathe. I think I lean towards the latter: The OP says it’s all carved from one piece of wood, and there’s no practical reason at all to do that, since whatever it’s for, it’d be easier to make it out of separate pieces.
You would have a smooth finish if the material could snag. It could also be polished from the work itself. If it is decorative, it is not a well finished decoration. The grain runs end to end. So it would not be too weak.
Holder for a pair of eyeglasses? 
Perhaps I didn’t make myself clear enough: A tool could certainly be polished through use. But it looks like that object has had some sort of glossy substance added to its surface, varnish or the like, to finish it, and that doesn’t seem likely for a tool. For a tool, you’d at most oil it, which wouldn’t look that glossy.
I do suspect it is a decorative piece. Though somewhat crude. But sure as the Sun orbits the Earth. First impressions can be wrong. My post was obviously worded to convey uncertainty. Just alternative possibilities. Tools can be decorative as well.
You might make them back to back like that - the pointed ends make it easier to get the parting tool in.
I was thinking this too! Maybe it for knitting woolen socks ???
The object is not worn smooth from use. It has some sort of finish on it.
My aunt had no children, so it’s not some object her child produced. Could possibly have been produced by one of her brothers when they were growing up in the 1920s-30s.
Actually that was precisely the sort of thing I was thinking of. Methodist lineage going back a very long way. A modern Methodist chapel is the last place you would find a Rood Screen, but back at the start when the overlap between CoE and Methodist was greater, I can imagine a forbearer taking something just like that as a memento.
I am going off down a deep rabbit hole here. Get it carbon dated and we can discuss some more
Lace bobbin? It looks like a combination of these two things:
Here’s a similar double ended acorn. It’s a needle case, apparently.
Here’s another.
also a similar object
and this:
Try seeing if the acorns unscrew.
Since it came from a desk, I’m guessing it was either a couple of lace bobbins or newel posts that were repurposed for rolling over pages turned over an ink blotter, back when people wrote with ink that needed to be blotted, which wasn’t that long ago, because even in the days of electric typewriters, there were many types of letters which decorum dictated should be handwritten.
My grandmother used a child’s toy rolling pin for this purpose, until my mother claimed her old toy for me when I was about three, so my grandmother used it for about 24 years. If anyone had found it, I’m sure they would have wondered why my grandmother had a tiny rolling pin in her desk.
What would help is a couple more pics. One with something to allow scaling it would be very useful. The suggestions provided so far range across a large set of possible sizes for the object. Also, a picture of the gap where there is roughly cut wood and no finish. So we can see the side of the square middle component and the edge of the finish on the middle square component.
I suspect the finish was applied after the object was assembled into a larger thing. Judging by the edge of the square bit it looks as if there may have been some trimming done just before fitting as well. The nature of the edge and the finish would give clues about the order of manufacture and its relationship with other bits.
It seems very unlikely, given the rough nature and lack of finish on the two central connection parts that this was a standalone object. It was part of a greater whole.
“about 6 inches end-to-end”
Right there in the first post.
