Brian Reay was thinking very hard :
I need to polish the ends of a few (well 4 bits) of short perspex rod, about
3mm in diameter after cutting. The bits of rod are to form 'light pipes' in a
display- each has an LED at one end.

Any ideas how best to polish the ends?

Flatten the ends first, with successively finer wet & dry, then polish
up after the finest with a fine slightly abrasive polish such as
toothpaste, Brasso, chrome polish, or similar.