Orthognathic surgery is jaw surgery that will reposition part of the jaw. It can include reshaping as well. It's usually done to solve severe problems like underbites that haven't responded well to non-surgical treatments like oral appliances, as well as in cases that are so severe that non-surgical treatments can't be done at all.
This surgery is usually bookended by orthodontic treatment. Pre-surgery treatment is meant to move teeth into position so that, after the surgery, there is little to no extra orthodontic treatment needed to make the patient's teeth match up when the jaws are closed together. Post-surgery treatment is meant to either make final adjustments to tooth position, or to retain the current tooth position with a retainer as the patient's jaw heals.
For people who have dealt with braces before or who simply don't want cemented brackets, a common question is if they can use a tray system like Invisalign instead. The answer is yes, but with qualifications.
The Orthodontist's Expertise in Invisalign Techniques
If you go to an orthodontist who doesn't usually use Invisalign, or who has just started offering the procedure, chances are you're not going to get Invisalign before or after the surgery simply because of the orthodontist's lack of experience with the product. If you want to use that type of treatment instead of braces, you absolutely need to find an orthodontist who has a lot of experience with tray systems and who has also worked with jaw surgery patients before, as planning orthodontic treatment around jaw surgery can be a delicate situation.
Communication Between the Surgeon and the Orthodontist
There also needs to be good, ongoing communication between the surgeon and orthodontist. If both are appropriately skilled in what the surgery and orthodontic treatment entail, they shouldn't have to be in constant contact, of course; what they should do, however, is agree that the overall general timeline for you is appropriate and to agree on how much the teeth need to be moved before surgery. The orthodontist also needs to know how the surgeon will change your jaw alignment so the tray treatments actually help.
The Success of the Surgery Itself
Even if you can use a tray treatment like Invisalign before the surgery, the success of the surgery itself may dictate whether you have to use braces afterward or can go back to a tray system. You may need the old bracket-and-wire system in the end if there were real problems with the surgery. There's no way to predict this, so you can't assume that if A or B happens, you have to go with C or D treatment. But the post-surgery success is a concern, so if complications occur, keep the orthodontist up to date and ask if you can still us a tray system if your teeth need more adjustment afterward.
If you find an orthodontist with a lot of Invisalign and jaw surgery experience, and work with a surgeon who is familiar with how Invisalign can be used with the surgery, you should be fine in general. Follow all instructions carefully to encourage the best results.