This starts on the ground. No 'spinning' necessary! :)
First, you want to be sure you can actually control the horse's shoulders and hips. It's so much easier to teach the side-pass when the horse can turn over its haunches and over its forehand when asked. Practice a few circles on the forehand (moving the haunches around so that the back legs are crossing over each other and the front end stays stationary) and on the haunches (rear stays stationary and front legs cross over). Once you've got those down pat, move to a wall or fence and ask front end, hind end, front end, hind end, until the horse is moving sideways smoothly. Keep in mind it's more difficult for your horse to bend in the direction it's sidepassing, so if they are a little crooked at first, it's alright.
Since the wall/fence is there to keep the horse from going forward, try not to tug on the halter anymore than you have do. I hold onto it and stand at the horse's side, facing the horse's rear, and barely nudge the horse's nose toward me as I ask it to step to the side.