I just installed a new toilet in my basement bathroom. While working around the bowl I accidentally twisted it. It seemed to pivot around the drain connection (so only the front part moved). I don't know if this has anything to do with the fact that the flange bolts may have moved out of their "locked" position in the flange or maybe something to do with the wax ring? The flange bolts don't sit perfectly vertical due to unlevel concrete floor under flange.
I'd appreciate any suggestions as to remedy this. I'd prefer not to seal the whole toilet to the floor.

The brass bolts that hold bowl down to the floor ( and closet ring ) should be first attached to the closet ring, toilet bowl set over them, and then tightened with second set of nuts. That way, bolts are firmly attached to the closet ring and won't move in the ring slot.

Aslo: if you have ABS ring with metal rim than the rim has to be firmly attached to the floor first, then the bolts go in second.

In any case, you should caulk around the base of the bowl with silicone caulk. Once dry, caulk will absorb any uneveness of the floor, provide soft cushion bet. Bowl and floor, as well as will prevent bowl from any movement.

Hey Milo, thanks for the response. Sorry about the incorrect terminology, I have the closet ring set up... but is it possible for this spinning issue if the bolts do not sit properly under the ring? It seems they don't fit as well as they should under the "locking" end of the ring.
Thanks again for your help!

If you say "don't fit" - what do you mean? Toilet bolts fit all toilet rings. If you think they are loose, than get a few washers and put one under the ring, one directly on top of the ring - then tighten it together with nut. Now, bolts should feel solid. Proceed with installing toilet bowl, another washer and another nut...

The problem with the bolts fitting is that the concrete floor underneath the ring is somewhat uneven. When you slide the bolt in it has room at the end for the head to be twisted to fit snuggly under the ring. I can't twist the head of the bolt to fit under this part therefore seeming to not give it as tight of as fit as it should have.

Take skinny chisel and chisel out concrete from under neath. You have to play with it. Then, insert bolt and twist it using channel lock. Also, take a hammer at tap the ring so any concrete attached to the rim will get loose and will fall off.

If you have ABS closet ring with metal rim than the rim will spin or even can be removed. See if you can remove it, clean under, and reinstall...