I want to have orthodontic surgery to move my maxilla forward. What is this procedure called and can it bring out my cheekbones?

I would definitely recommend double jaw surgery. Based on the x-ray, the upper jaw is set back in relation to the nose and the cranial base. In addition, any horizontal discrepancies greater than 7-8 mm should be performed in 2 jaws as a general rule of thumb. This is due to long-term statistics...

It appears that your mandible is definitely longer on one side, causing an asymmetry. I am not sure if there are any TMJ problems other than popping (like pain and episodes of locking). Now, as part of the full informed consent process prior to initiation of orhtodontia, you should have been...

 I think that your best option would be orthodonticswith bicuspid extractions as well as a chin reduction  Genioplasty.   This will take care of the bimaxillary protrusion as well as the strong chin   With a reduction genioplasty, your lower jaw will be softened...

 My advice to you is to wait 6 to 12 months prior to making any decisions about reversing the surgery.   Yes, the surgery can be reversed. However, the last 15 to 20% of swelling can take several months more to finally resolve.  Dr. David Mueller, DDS, FABOMS

This is a great question. In general it depends on how well the plates hold the bones together. In my practice we will usually place rubber band and try to plan for the plates and screws to be placed in strong bone. 

Based on your photo, the surgical movements I would recommend would be a 60-70% maxillary advancement and a 30-40% mandibular set back. I think the main issue is midface deficiency, so by bringing the upper jaw forward, your cheeks should look more full. The lower jaw is somewhat forward, but...