Swirls and Scratches on Glossy Back Wood Surface

There's cleaner/wax for piano black available just for this. Check music stores. I'd start with this first.

There's plenty of automotive products available for varying degrees of damage, but in most cases they're far to abrasive for what you're looking for. I don't think you need much to bring back a mirror finish.

As mentioned, Novus makes a great product as well, #1 is a good all around product. I'm getting some piano black speakers soon, and am also getting a car duster (micro fiber fingers), a mitt, and piano black cleaner/wax. Should never have any problems.
I'm a motorcycle buff, but don't think you need any strong products for something that's never seen bug splatters or mud. There's something out there for almost every issue though.

A microfiber cloth is perfect for polishes . recently had to clean up some piano black finished speakers to sell . 3m polish and a microfiber worked great on fine swirl marks. All done by hand 

Guitar makers like Collings use Novus so there's that...nobody makes wood finishes like high end guitar companies, and I was turned on to the stuff by Pat Di Burro of Essex, NH, who has worked on some of my guitars and is considered the premier guitar repair expert in the North East and one of the best in the country (official repair guy for many high end companies). Use the lightest (#1) by hand.

To apply 3M Perfect-it, I need to use a buffer polisher, right? Or, Is it possible to apply it by hand?

Novus is designed for plastics like Lexan and it works great for aircraft windscreens. I'd proceed with caution if trying it on a speaker finish. 

3M Perfect-It products are the best I've used for automotive type finishes which many black speakers have.

Novus 1 is still worth a try.

ozzy

Thank for the responses.

I just checked the Novus, but it says it's for plastic. Does it work for wood too? @ozzy

Try the Novus 1,2,3 polish

ozzy

All too common a problem with glossy black finish. Some of the so called swirl removers do nothing more than hide the scratches. A friend told me once, if you like shiny black cars, take a good look at it when new, cuzz it will only get worse looking as time goes on....

I'd try a wax or polish first, one specifically designed for piano finishes. If that doesn't do the trick, you can try something more aggressive like 3M Perfect-It swirl remover. Make sure to first test any product on the bottom of the speaker.