The actual cap has two small metal projections sticking into the wood to keep the cap from rotating. Be sure to NOT twist the cap until you get the cap UP and loose or you can chip the wood.
As mentioned the factory can supply you with new medallions. The shipping/handleing is actually more than the cost of a single medallion. So I usually buy 5 or 6 at a time just to make it worth while. I use a just SLIGHTLY undersized hardwood dowel about 4 inches long to tap/drift the medallion into place.
The other alternative to prying from inside the hole you drill in the medallion, is to drill the hole centered and just large enough to get a phillips screwdriver in the hole. Then back out the screw and the screw should bring the medallion up and out as is comes out.
RWT