By "invasive" people probably mean it is an aggressive spreader in a garden setting, like many mint family plants are. As far as I know, all, or nearly all, Monarda species are native to the U.S.

To add to what G-Gal has said, the very fact you are concerned about this sun-loving plant spreading into surrounding forest is likely a sign of your good intentions, but is likely not worth worrying about. True forest invaders are almost always shade-tolerant. This Monarda? No way.

I have two annual monardas in my field. Monarda citriodora and Monarda punctata. They are spreading, by seeding out in my field. I love them but this is not a garden in a traditional sense. I have two paper bags on my porch waiting for me to sow out in my front field. I collected the M. punctata from the Tent Rocks canyon in New Mexico. It is native to my county in Texas, but I have not seen it around . IT has proven to be very happy here. I do have to yank it out in the spring where I do not want it , mostly around my small cactus.. It loves that sandy soil.

I find that the concept of the "mixed border" is constraining to many species that naturally want to make a sizable clump. Some plants need quite a bit of real-estate to be happy. My memory of the clump of a perennial monarda is one of them. I would not want to call it a garden thug, just a plant with big feet. I removed it out of my wet zone because it was too big for that area. and too much of a water lover for my dry hill. Monarda fistula was another more East Texas native who did not like my thin soil. It took a could of years to get all the roots from around the grey water pipe but it I did. It would grow but not bloom because of lack of sun under the oaks.

I love monarda, for both it's aesthetic beauty, and for it's popularity with pollinators. M. punctata always attracts the greatest diversity of insects, and M. citriodora reminds me of some plant from a Dr. Seuss book. I have been ill the past few weeks, so haven't gotten out to the garden much, but my husband noticed a large black butterfly and I have at least seen how floriferous my M. fistulosa is this year, now that it is better established. I also have a darker, magenta Monarda that I don't have a name for, that has also spread well this year. Eventually, I will have a relatively low maintenance front bed of native pollinator plants.

Martha

I forgot to mention that they have a tendency to mildew in the shade. Another reason for no bloom.

Should have said at the outset-the choice is not limited to native or invasive! many shades of gray inbetween. Many non-native plants have naturalized here in N. America. That is not necessarily a bad thing. Just to cite one example, the common roadside weed chicory is naturalized all over the place. It is not however capable of pushing out whole native plant communities, so is not invasive, or not a top-tier invasive. Stuff like that. A plant can be both non-native and non-invasive. It is true however that being able to naturalize is a prerequisite for becoming invasive.