This past week I started this swirl removing process. I wanted to go with the least abrasive polish so I started with Meguiar's Ultimate Polish.. and if that didn't work I was going to go over to the compound.. but as you can see, it worked quite well!
In the 3rd photo, it looks like there are swirls left but those are actually reflections of the sun off the garage door above the car.
There are still a decent amount of microscratches if I hold a halogen light above the paint, but those are of course much less noticeable than the insane swirls I had before.
So depending on how bad your swirls are, mine looked pretty bad.. and the ultimate polish did the trick and I hear the ultimate polish hardly removes any of the clearcoat.