Testing the Addition of Chemotherapy or Chemo-Immunotherapy to the Usual Surgery for Advanced Head and Neck Cancer

Inclusion Criteria: * Pathologically (histologically or cytologically) proven diagnosis of locally recurrent or persistent squamous cell carcinoma of head and neck (SCCHN) arising within the oral cavity, oropharynx, larynx, or hypopharynx * PD-L1 combined positive score (CPS) ≥ 1 using a Clinical Laboratory Improvement Amendments (CLIA) certified laboratory * Verify insurance (or other payment) coverage for neoadjuvant chemotherapy * Measurable disease as defined by RECIST 1.1 * Patients must have locally recurrent or persistent SCCHN arising within the oral cavity, oropharynx, larynx, or hypopharynx (American Joint Committee on Cancer \[AJCC\] Cancer Staging Manual, 8th Edition) AND are deemed candidates for salvage surgery: * P16 positive oropharynx patients with T2, T3, T4, N0, N1, N2 and all other patients with T2, T3, T4a, N0, N1, N2a, N2b, N2c, N3a are eligible. * Patients must be deemed surgically resectable without gross residual disease. * For patients with oral cavity SCCHN, only those with recurrent or persistent disease after prior surgery are eligible. * Patients who are candidates for salvage laryngectomy to treat recurrent laryngeal cancer and who are having salvage surgery for curative intent are eligible. * Patients with resectable lymph node-only recurrence are eligible. * No major vascular involvement (\> 180° involvement of the common carotid or internal carotid artery), jugular foramen involvement, or prevertebral, paraspinous muscle involvement precluding a curative resection * No evidence of distant metastatic disease * The following minimum diagnostic workup is required: * General history and physical examination. * Diagnostic-quality neck CT and PET/CT of neck (PET with attenuation-correction CT of neck, chest, and abdomen) * Age ≥ 18 * Eastern Cooperative Oncology Group (ECOG) performance status of 0-2 * Negative urine or serum pregnancy test (in persons of childbearing potential) within 30 days prior to registration. Childbearing potential is defined as any person who has experienced menarche and who has not undergone surgical sterilization (hysterectomy or bilateral oophorectomy) or who is not postmenopausal * Absolute neutrophil count (ANC) ≥ 1,500 cells/mm\^3 * Platelets ≥ 100,000 cells/mm\^3 * Hemoglobin ≥ 8.0 g/dL (Note: The use of transfusion or other intervention to achieve hemoglobin \[Hgb\] ≥ 8.0 g/dL is acceptable) * Adequate renal function defined as creatinine clearance (CrCL) \> 50 mL/min by the Cockcroft-Gault formula * Total bilirubin ≤ 1.5 x institutional upper limit of normal (ULN) (Not applicable to patients with known Gilbert's syndrome) * Aspartate aminotransferase (AST)(serum glutamic oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT) (serum glutamic pyruvic transaminase \[SGPT\]) ≤ 3 x institutional ULN * Only patients who received prior radiation therapy in the definitive or post-operative setting (limited to one course) are eligible. * Prior radiation therapy must have been completed at least 6 months prior to registration with the majority of the index persistent/recurrent cancer volume (\> 50%) irradiated to ≥ 40 Gy at the time * Prior systemic therapy including immunotherapy with anti-PD1 or anti-PDL1 within the definitive setting (neo-adjuvant, or adjuvant) is permitted and must have been completed at least 4 months prior to registration * Prior systemic therapy including immunotherapy for treatment of recurrent or metastatic SCCHN is not permitted * No investigational anti-cancer agents received within 4 weeks prior to registration * No New York Heart Association Functional Classification III or IV (Note: Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification) * No active infection requiring IV antibiotics, IV antiviral, or IV antifungal treatments * No peripheral neuropathy grade 3 or 4 * No history of interstitial lung disease * No active, noninfectious pneumonitis requiring immunosuppressive therapy * No history of a solid organ transplant (other than corneal transplant) * No active autoimmune disease that has required systemic treatment in the past 2 years (i.e., use of disease-modifying agents, corticosteroids \[\> 10 mg prednisone/day or equivalent\] or immunosuppressive drugs) * NOTE: Patients meeting the following criteria are not considered immunosuppressed and are eligible to enroll: * Patients who require a brief course of steroids (e.g., prophylaxis for imaging assessments due to hypersensitivity to contrast agents) are not excluded * Patients with type I diabetes mellitus, and endocrinopathies (including hypothyroidism due to autoimmune thyroiditis) only requiring hormone replacement, or skin disorders (such as vitiligo, psoriasis, or alopecia) not requiring systemic treatment, or conditions not expected to recur in the absence of an external trigger are permitted to enroll * Physiologic replacement doses ≤ 10 mg prednisone/day or equivalent are allowed. Inhaled or topical steroids are permitted * No history of allergic reaction to the study agent, compounds of similar chemical or biologic composition to the study agent, and immune checkpoint inhibitors (or any of its excipients)