Inclusion Criteria:

          1. Cytologic or histologic diagnosis of squamous cell carcinoma of the oral cavity,
             p16-negative oropharynx, hypopharynx, or larynx.

          2. Tumors must be HPV-negative. For eligibility, tumors of the oral cavity, hypopharynx,
             or larynx will be considered HPV-negative without specialized testing. Tumors of the
             oropharynx must be HPV-negative as determined by p16 immunohistochemistry and/or
             HPV-DNA per local standard.

          3. Clinical stage I-IVa based upon the American Joint Committee on Cancer (AJCC) staging
             manual, 8th edition.

          4. Appropriate and planned for oncologic resection of the primary tumor and/or neck
             dissection.

          5. Clinically or radiologically measurable disease; the primary tumor and/or cervical
             nodes may be measurable according to RECIST 1.1 (tumor diameter ≥ 1 cm; short-axis
             lymph node diameter ≥ 1.5 cm) OR by caliper/ruler measurement (tumor diameter ≥ 1 cm).

          6. No prior treatment for the index (study-qualifying) HNSCC.

          7. Patients with two simultaneous primary tumors or bilateral tumors are included.

          8. The index HNSCC may be a second primary HNSCC, provided the following criteria are
             met:

             a. The previously treated HNSCC was treated with curative intent. b. The index HNSCC
             is at least 1 cm from the previously treated HNSCC. c. At least 2 years have elapsed
             since curative treatment of the previous HNSCC without evidence for recurrence.

          9. Eastern Cooperative Oncology Group (ECOG) performance status 0-1. (See Appendix 1.)

         10. Adequate hematologic function, as defined by:

             a. Absolute neutrophil count (ANC) ≥ 1,500/µl b. Platelets ≥ 100,000/µl c. Hemoglobin
             ≥ 8 g/dL

         11. Adequate liver function, as defined by:

             a. Bilirubin ≤ 1.5 x institutional upper limit of normal (ULN). Patients with
             Gilbert's syndrome with a total bilirubin ≤ 2.0 x ULN and direct bilirubin within
             normal limits are permitted.

             b. aspartate aminotransferase (AST) and alanine aminotransferase (ALT) ≤ 3 x ULN

         12. Adequate renal function, as defined by creatinine ≤ 1.5 x ULN.

         13. Able to swallow oral medications.

         14. Have signed written informed consent.

         15. Consent to biomarker collection requirements, including mandatory baseline and
             intra-operative research biopsies of the index tumor.

        Exclusion Criteria:

          1. Prior treatment with any cyclin-dependent kinase (CDK) 4/6 inhibitor or
             anti-programmed death (PD)-1/L1 inhibitor is not allowed.

          2. Current or prior use of immunosuppressive medication within 14 days before the first
             dose of their assigned protocol treatment. The following are exceptions to this
             criterion unless otherwise indicated:

               1. Intranasal, inhaled, or topical steroids, or local steroid injections (eg, intra
                  articular injection)

               2. Systemic corticosteroids at physiologic doses not to exceed 10 mg/day of
                  prednisone or its equivalent

               3. Steroids as premedication for hypersensitivity reactions (eg, CT premedication)
                  and/or as anti-emetics

          3. Active or previously documented autoimmune or inflammatory disorders (including
             inflammatory bowel disease [eg, colitis, Crohn's disease], diverticulitis [with the
             exception of diverticulosis], systemic lupus erythematosus, sarcoidosis syndrome,
             Wegener syndrome [granulomatosis with polyangiitis], Graves' disease, rheumatoid
             arthritis, hypophysitis, uveitis, etc.). The following are exceptions to this
             criterion:

               1. Patients with vitiligo or alopecia

               2. Patients with hypothyroidism (eg, following Hashimoto syndrome) stable on hormone
                  replacement. Subclinical hypothyroidism (eg. Elevated thyroid-stimulating hormone
                  (TSH), low or normal free T4, and asymptomatic) observed on screening labs is not
                  an exclusion.

               3. Any chronic skin condition that does not require systemic therapy

               4. Patients without active disease in the last 5 years may be included but only
                  after consultation with the study PI

               5. Patients with celiac disease controlled by diet alone

          4. Patient has a personal history of any of the following cardiac or pulmonary
             abnormalities:

               1. Syncope of cardiovascular etiology

               2. Ventricular arrhythmia of pathologic origin (including, but limited to,
                  ventricular tachycardia and ventricular fibrillation)

               3. Sudden cardiac arrest

               4. Documented history of New York Heart Association functional classification III-IV
                  congestive heart failure

               5. Myocardial infarction ≤ 6 months prior to enrollment

               6. Current unstable angina pectoris

               7. Interstitial lung disease

               8. Severe dyspnea at rest or requiring oxygen therapy

          5. Patient with impaired gastrointestinal (GI) function or GI disease that may
             significantly alter the absorption of oral abemaciclib (e.g. history of major surgical
             resection involving the stomach or small bowel, malabsorption syndrome, preexisting
             Crohn's disease or ulcerative colitis, preexisting chronic condition resulting in
             baseline Grade 2 or higher diarrhea).

          6. Patients who require the chronic administration of drugs that are strong and moderate
             inducers of Cytochrome P450, family 3, subfamily A, (CYP3A) and/or strong inhibitors
             of CYP3A, and no acceptable substitute can be identified, are not eligible for study
             (See Appendix 2). Such drugs should be discontinued at least 7 days before the start
             of study treatment.

          7. The patient has active bacterial infection (requiring intravenous [IV] antibiotics at
             time of initiating study treatment), fungal infection, or detectable viral infection
             (such as known human immunodeficiency virus positivity or known active hepatitis B or
             C [e.g. hepatitis B surface antigen positive or hepatitis C antibody positive with
             detectable viral load]). Screening for HIV or hepatitis is not required for
             enrollment.

          8. Patient with any other condition that would, in the Investigator's judgment, preclude
             patient's participation in the clinical study due to safety concerns or compliance
             with clinical study procedures, e.g. social/psychological complications.

          9. Pregnant or nursing (lactating) women.

         10. Patient who does not apply highly effective contraception during the study and through
             the duration as defined below after the final dose of study treatment:

               1. Sexually active males should use a condom during intercourse while taking
                  abemaciclib and for 4 weeks after the final dose of abemaciclib.

               2. Males who are sexually active with a woman of child-bearing potential should
                  apply highly effective contraception during the study as defined below, in order
                  not to father a child in this period.

               3. Women of child-bearing potential (WOCBP), defined as all women physiologically
                  capable of becoming pregnant, must use highly effective contraception during the
                  study and through at least 3 weeks after the final dose of abemaciclib (Arm 1),
                  and 20 weeks after the final dose of nivolumab (Arm 2). Highly effective
                  contraception is defined as:

             i. Total abstinence: When this is in line with the preferred and usual lifestyle of
             the subject. [Periodic abstinence (e.g., calendar, ovulation, symptothermal,
             postovulation methods) and withdrawal are not acceptable methods of contraception].

        ii. Female sterilization: have had surgical bilateral oophorectomy (with or without
        hysterectomy) or tubal ligation at least six weeks before taking study treatment.

        iii. Male partner sterilization (with the appropriate post-vasectomy documentation of the
        absence of sperm in the ejaculate). [For female study subjects, the vasectomized male
        partner should be the sole partner for that patient.] iv. Use a combination of the
        following (both 1+2):

        1. Placement of an intrauterine device (IUD) or intrauterine system (IUS) 2. Barrier
        methods of contraception: Condom or Occlusive cap (diaphragm or cervical/vault caps) with
        spermicidal foam/gel/film/cream/vaginal suppository.

        v. Note: Hormonal contraception methods (e.g. oral, injected, and implanted) are not
        allowed as abemaciclib may decrease the effectiveness of hormonal contraceptives.

        vi. Note: Women are considered post-menopausal and not of child-bearing potential if they
        have had 12 months of natural (spontaneous) amenorrhea with an appropriate clinical profile
        (e.g. age appropriate, history of vasomotor symptoms) or have had surgical bilateral
        oophorectomy (with or without hysterectomy) at least six weeks ago.

        11. Prisoners or subjects who are compulsorily detained (involuntarily incarcerated) for
        treatment of either a psychiatric or physical (e.g., infectious) illness.