Transoral Surgery Followed By Low-Dose or Standard-Dose Radiation Therapy With or Without Chemotherapy in Treating Patients With HPV Positive Stage III-IVA Oropharyngeal Cancer

Inclusion Criteria: Registration to Surgery (Arm S) * Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 * Patients must have newly diagnosed, histologically or cytologically confirmed squamous cell carcinoma or undifferentiated carcinoma of the oropharynx; patients must have been determined to have resectable oropharyngeal disease; patients with primary tumor or nodal metastasis fixed to the carotid artery, skull base or cervical spine are not eligible * Patients must have American Joint Committee on Cancer (AJCC) TNM tumor stage III, IV a, or IV b (with no evidence of distant metastases) as determined by imaging studies (performed \< 30 days prior to pre-registration) and complete head and neck exam; the following imaging is required: computed tomography (CT) scan with IV contrast or magnetic resonance imaging (MRI) * Patients must have biopsy-proven p16+ oropharynx cancer; the histologic evidence of invasive squamous cell carcinoma may have been obtained from the primary tumor or metastatic lymph node. It is required that patients have a positive p16 IHC (as surrogate for HPV) status from either the primary tumor or metastatic lymph node. * Carcinoma of the oropharynx associated with HPV as determined by p16 protein expression using immunohistochemistry (IHC) performed by a Clinical Laboratory Improvement Amendments (CLIA) approved laboratory; using p16 antibody obtained from Roche mtm laboratories AG (CINtec, clone E6H4) is recommended * Patients wit…