Los Tres Paso: Neoadjuvant Palbociclib Monotherapy, Concurrent Chemoradiation Therapy, Adjuvant Palbociclib Monotherapy in Patients With p16INK4a Negative, HPV-Unrelated Head and Neck Squamous Cell Carcinoma

Inclusion Criteria: * Larynx SCC, hypopharynx SCC, or oral cavity SCC. HPV-unrelated OPSCC \[defined as p16INK4a negative by IHC (staining in \< 70% of cells) or HPV High Risk (Type 16 or 18) negative by ISH\]. P16INK4a positive larynx SCC, hypopharynx SCC, and oral cavity SCC are eligible given the unknown effect of this on the biology of SCC of these subsites. * Overall Stage III, IVA, or IVB disease per AJCC version 7.0 * Measurable disease defined as lesions that can be accurately measured in at least one dimension (longest diameter to be recorded) as ≥ 10 mm with CT scan, as ≥ 20 mm by chest x-ray, or ≥ 10 mm with calipers by clinical exam. * At least 18 years of age. * Normal bone marrow function as defined below: * Absolute neutrophil count ≥ 1,000/mcL * Platelets ≥ 100,000/mcL * Hemoglobin ≥ 9.0 g/dL * QTc \< 500 msec by Fridericia * Women of childbearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control, abstinence) prior to study entry, for the duration of study participation, and for 90 days after completion of treatment. Should a woman become pregnant or suspect she is pregnant while participating in this study, she must inform her treating physician immediately. * Ability to understand and willingness to sign an IRB approved written informed consent document (or that of legally authorized representative, if applicable). Additional Cohort 1 Eligibility Criteria: Patients enrolling to Cohort 1 must meet all of the following criteria: * Eastern Cooperative Oncology Group (ECOG) performance status 0-1 * Adequate organ function defined as: * Serum creatinine ≤ 1.5 x institutional upper limit of normal (IULN) and creatinine clearance ≥ 75 mL/min * Bilirubin ≤ 1.5 x IULN * ALT and AST ≤ 2.5 x IULN Additional Cohort 2 Eligibility Criteria: Patients enrolling to Cohort 2 must meet at least one of the following criteria: * ECOG performance status of 2 * Reduced organ function defined as: * Creatinine clearance 30-75 mL/min * Bilirubin 1.5-2 x IULN * ALT and AST 2.5-5 x IULN Exclusion Criteria: * Diagnosis of cutaneous, paranasal sinus, salivary, or nasopharynx SCC, or diagnosis of neck nodes with unknown primary. * Diagnosis of P16/HPV-ISH positive OPSCC. * Presence of distant metastatic disease. * Prior systemic therapy for current diagnosis of HNSCC. * A history of other malignancy ≤ 2 years previous with the exception of basal cell or squamous cell carcinoma of the skin which were treated with local resection only or low risk/curatively treated prostate, thyroid, and cervical cancers. * Currently receiving any other investigational agents. * Treated within the last 7 days prior to Day 1 of protocol therapy with: * Food or drugs that are known to be STRONG CYP3A4 inhibitors (e.g. grapefruit juice, verapamil, ketoconazole, miconazole, itraconazole, erythromycin, clarithromycin, telithromycin, indinavir, ritonavir, nelfinavir, atazanavir, amprenavir, nefazodone, diltiazem, and delavirdine) or inducers (e.g. glucocorticoids, progesterone, rifampin, phenobarbital, St. John's wort) \[moderate CYP3A4 inhibitors/inducers are okay\] * Drugs that are known to prolong the QT interval * Drugs that are proton pump inhibitors * A history of allergic reactions attributed to compounds of similar chemical or biologic composition to palbociclib, cisplatin (for Cohort 1), or cetuximab (for Cohort 2). * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or uncontrolled electrolyte disorders that can compound the effects of a QTc-prolonging drug (e.g. hypocalcemia, hypokalemia, hypomagnesemia). * History of cirrhosis. * History of renal or liver transplant. * Pregnant and/or breastfeeding. Women of childbearing potential must have a negative serum pregnancy test within 28 days of study entry. Female patients must be surgically sterile or be postmenopausal, or must agree to use effective contraception during the period of the trial and for at least 90 days after completion of treatment. * Known HIV-positivity and on combination antiretroviral therapy because of the potential for pharmacokinetic interactions with palbociclib. In addition, these patients are at increased risk of lethal infections when treated with marrow-suppressive therapy. * Rb (retinoblastoma) loss: mutation or homozygous deletion described on genomic sequencing report.