I'm aware that the spinner can do that, so I do two things to prevent it from happening:

a) Right before the spinner hits the water on the cast, I will use my thumb to slow the spinner down before it hits the water. This causes the entire spinner and barrel swivels to straighten out as it splashes in the water.

b) I give the spinner a quick hard jerk as soon as it disappears in the water. This way, I can instantly feel if it's working or not.

It can happen, but it doesn't happen a lot. Not enough to remove the barrel swivel, IMO.
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T.K. Paker