What Sam said.
It's a common ( and potentially fatal in this case) error to confuse shiny with smooth and flat.
It would be a piece of cake to hit your HS with a buffing wheel and make it shiny, but if it's not flat and (relatively) scratch free then you've accomplished nothing.
If you were referring to the finned areas of the HS then that's a different story.
Copper stains/oxidizes very easily and the shape of the HS makes it very difficult to polish out, although it could probably be done.
A good cleaning in a ultasonic bath with the proper solution would be about the most painless method I can think of.
Most modern clock repair shops have exactly the setup required to do this, perhaps you could see if a local shop would be willing to clean yours for you.
That would get you as clean as possible.
Depending on the design of the HS, tumbling it would also achieve a very nice high polish, but I know that my Zalman wouldn't survive the process...the fins are way too thin and fragile.