Hhmmm....
Normally, once you've wound your backing and fly line on to your reel, there is enough pressure to keep the whole thing from rotating (if I understand correctly what happened). What did you use for backing? Sometimes, the new "superbraid" lines are very slick and this might allow for slippage around the spool.
I'd suggest you strip out in the backyard your entire fly fline and backing. Re-tie the backing to the spool, reel four or five turns of backing on to the spool. Then get some duct tape and tape the spool covering over the four/five wraps (try to cover the whole spool). Once the tape is on, then continue to wind backing, making sure that the first few winds over the tape are pretty firm - after a few winds you should be able to ease off on tension using just enough to keep the backing and fly line level and even on the reel.
This should fix your problem.