It would be similar to removing swirls on automotive paint. Swirls are tiny scratches in the paint. You're not removing them with something you just spray on and wipe around. Removing swirls from paint involves a product with microscopic abrasives and takes elbow grease to scrub at the paint. Essentially, you are not removing swirls. You are removing the paint with abrasives around the swirls to get it down to a level that the swirls are at.

Note, you're bicycle's paint is probably a lot softer than the hard clear coat of an automobile. Meguiars Swirl-X would probably be good for that. Meguiars Scratch-X is also a great product, but a little more abrasive than Swirl-X.