Nivolumab & IRX-2 With Surgery for Resectable Stage III-IVA Oral Cavity Cancer or HPV-Positive Oropharyngeal Cancer

PRIMARY OBJECTIVES: I. To determine the safety profile of combination immunotherapy, nivolumab + IRX-2, for HPV+ oropharyngeal squamous cell carcinoma (OPSCC) and HPV- oral cavity squamous cell carcinoma (OCSCC). II. To assess the oncologic efficacy of neo-adjuvant immunotherapy using pathologic confirmation of response after surgical resection. SECONDARY OBJECTIVES: I. To correlate tumor microenvironment histopathology with pathologic findings, with progression free survival (PFS) and other outcome parameters in patients with resectable OPSCC and OCSCC after the above treatments. II. To evaluate swallowing function before and after surgery and risk-adjusted adjuvant therapy. III. To evaluate quality of life (QOL), swallowing perception and performance, voice outcomes, and head and neck symptoms. OUTLINE: Participants receive nivolumab intravenously (IV) over 60 minutes on days 1 and 15, cyclophosphamide IV on day 1, and IRX-2 subcutaneously (SC) over 10 consecutive days between days 4-21 in the absence of disease progression or unacceptable toxicity. Beginning days 25-30, participants undergo surgery. After completion of study treatment, patients are followed up at 3 months, every 3 months for 2 years, then every 6 months for 2 years.