I use a chiropractor vet, and he and my now retired vet both are big fans of massage, done by someone good. The local veterinary board was trying to get massage therapists here have to work with vets or be fined for practicing veterinary medicine without a license - both vets spoke up in favor of our massage therapists.
My gelding had an SI injury after a bad wipeout in turnout, and he got massage before and after the vet adjusted him, because his muscles would spasm when he was realigned and the massage helped get them to relax and not immediately pull him out of whack again. A few months of this and he was in great shape.
Really well ridden dressage helps eliminate the need for either in many cases, but if my trainer weren’t riding and helping fix crookedness and tight areas I created we’d look at more chiro and massage for my horses, because when I’m riding it can’t be called really well ridden dressage!