Paint the ceiling the white of your trim, but do it in a flat finish.

Textured ceilings aren't painted when the texture is applied. It comes down very easy if the texture gets too wet. When painting ceilings you need to use a deep nap roller and move quickly. Once the first coat is on the second is easier as the texture is sealed better. Flat ceilings are all the rage but they are very labor intensive as you need to hide sooo much perfectly. There's nothing wrong with painitng any ceiling. What will make the ceiling stand out is if the walls are freshly painted and the ceiling is not.

Can you post a photo of your space? Do you mean you want to paint your ceilings a color other than white?

Do your ceilings have an Applied texture though, or are they plaster or spackle which has been textured when it was applied? Swirly sanded ceiling were very popular where I grew up, but it's part of the finish coat, it's not applied and it won't come off.

I would just paint the ceilings a flat ceiling white or the white of the crown molding if you have that.

Don't be ashamed of your swirl finish. It does not mean your home is cheap or that less care was taken in the finishing work. The repulsion that many Houzzers feel at the concept of textured walls and ceilings is misguided; in my region it is standard in higher end homes of a pre-mcmansion vintage.

I had a light blue textured ceiling in a previous home. I liked it. It's just ceiling paint. And it's YOUR ceiling paint. You have swirls. I bet many of your friends do as well and don't think less of you because of it. Go for it!

Swirl ceiling or walls is an upgrade. It takes a person with training and skill to do swirls. It is not cheap to have done. . That is why you don't see it done often presently. It is much cheaper and requires little skill to apply an overall sprayed texture to walls or ceilings such as orange peal. Just paint as you normally any other ceiling.

I had textured ceilings in my last house. Through the years all were painted, but, since it was an applied on texture, I did prime the ceilings first with oil based primer. That seals the texture, and you can then use latex paint to paint them with.