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Any tidally locked satellite such as the moon.

A tidally locked satellite rotates: one rotation per orbit.

unless the body it rotates around rotates at the exact same speed around another body in the opposite direction

Then it still rotates.

It can only rotate relative to an arbitrary point, not in some objective absolute sense. Here it does not rotate relative to chosen point.

Something about privileged reference frames...

But what's the alternative, right?