Yeah you can. I just put mine it my stainless tumbler when I got my Ross break. Now I did it when it was still raw metal, so I didn't need to worry about finish. So that would be the only thing you may need to consider.
The reason I put mine in the tumbler was to take out any burrs and to smooth everything out. Also to note, if this is a new break. I would either Time it after you tumble it or only time it to about 11:30. I timed mine prior to tumbling at 11:30 and after it tumbled, it timed perfect at 12.