To my knowledge, you cannot refinish Twist or Damascus "Finish" barrels. At least not and keep the pattern. They are not real pattern-welded barrels, they are steel barrels with a coating applied to make them look like the more expensive pattern-welded ones. Typically used on less expensive guns of Belgian origin. Look for Belgian proofs, as the Birmingham made types are actually of better quality and MAY be worth consideration. Birmingham gun's will have real pattern-welded barrels, though. I usually classify these types of guns, (Belgian) often called "Spurious" as they are named similar to more expensive guns to trick the unwary, as "Lamp Guns". They're not worth the time, money, and effort of restoration, but can be converted into nice Vintage decor items.

Some examples of name pairs, remembering that trademark law was weak or nonexistent at the time:

W Richards= Westley Richards
T Barker= Parker Bros
Sam Holt= Colt
A Greener= WW Greener

There are many more examples out there.

Here's Greener's explanation:
http://firearmshistory.blogspot.com/2012/06/spurious-guns-1.html
Regards
Ken

I prefer wood to plastic, leather to nylon, waxed cotton to Gore-Tex, and split bamboo to graphite.