We have a unique opportunity to get some used granite countertops at a steal of a price. Problem is, they are a mirror image of what would fit in our kitchen. Is there any reason why we wouldn't be able to flip them over and polish the underside and use it as the top? Any idea how much having it professionally polished would cost? (We could take them to the polisher, it wouldn't have to be an on-site job.) I'm just looking for a rough idea to see if it's even feasible. Granite would normally be out of our price range.

not sure about cost, but the risk of the tops cracking with all that additional handling would be high. You would still be on hook if they cracked.
I think your better off purchasing new and forget about the cost of polish. But if your still interested, go to the local granite guys and ask. Bet they will not be willing to do this. Also what about the edges? Are they Bull nose, or some sort of Ogee? Bull nose can be turned over, the rest, nope.

A decent granite shop should not have a problem polishing the underside of the granite, and it should be less expensive than new slabs, however there are other issues that may prohibit this. Depending on the type of granite (or if there is a cutout in the top), the tops could have metal rods cut into it on the underside for strength. If this is the case they would not be usable in this manner. Also like Home Care Club mentions, the edge profile could also prohibit the reuse. If they were able to be reused as you wish, in order to install the tops, the original polished face would have to be roughed up for the adhesive to bond them to the cabinetry. So there are a few things to consider before taking this one on. I would do as Home Care Club mentions and get the granite shop representative to look at them and quote polishing the underside AND installing them. Breakage would still be a risk YOU would assume and not the granite shop.

I agree with HomeOptions.