Yes you can put it back together I think they are just pressed together maybe with a little heat on the one side to expand the craters that the nipples fit in.

There's a thread somewhere here where someone makes a nine ball spinner out of a regular one with pics for reference.

I maintain a GP that is currently broken, but often available for public play and it has gone through 4 spinners in the past 3 years. They either separate or break off where the wire meets the metal plate.