Did you check whether the dimensions were the same side to side as top to bottom? I would expect them to be so, so it would not matter which way you installed it.
If I needed machining work done, and I found out that this is as close to .001" as your shop can even _measure_, I'd RUN, not walk, to the nearest exit!
The handle doesn't have a hole for a screw, it would make a good weapon but I don't know it was meant for that.
I measured it by rolling up a piece of paper, inserting it into the handle, taping the paper and then getting the diameter of the cylinder, which was
19/32". That makes it just under 5/8", probably thick enough for a fancy cane that's not meant to support much weight, I'm not familiar with riding crops and swagger-sticks so I don't know what size they usually were.
OK, then. I was going by the apparent ratio of opening diameter to length in the picture, and your stated length of 3.5 inches. Yes, 5/8 is about right for an evening cane.
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