Ibrutinib and Palbociclib in Treating Patients With Previously Treated Mantle Cell Lymphoma

Inclusion Criteria:

* Patients must have histologically or cytologically confirmed mantle cell lymphoma as defined by the World Health Organization; all patients must have either t(11;14) by karyotype or fluorescent in-situ hybridization (FISH) or positive immunohistochemistry for cyclin D1
* Subjects must have measurable disease defined as at least one tumor lesion of at least 1.5 cm or a peripheral blood CD5+, CD19+ lymphocyte count of at least 5,000 cells/uL
* Subjects must have received at least one prior treatment regimen

  * Subjects that have received a prior BTK inhibitor or CDK4/6 inhibition are ineligible
  * Subjects that have undergone prior allogeneic stem cell transplantation will only be eligible if the transplant occurred at least 1 year prior to study entry, the patient is no longer taking any immunosuppressive therapy, and there are no significant ongoing transplant-related adverse effects
  * Subjects must not have received chemotherapy =\< 21 days prior to first administration of study treatment, monoclonal antibody =\< 6 weeks prior to first administration of study treatment, and/or radiotherapy or other investigational agents =\< 4 weeks prior to first administration study treatment unless the subjects' tumor has progressed on the previous therapy and the investigator believes that the patient should not postpone further therapy and, all treatment-related toxicities have resolved to Common Terminology Criteria for Adverse Events (CTCAE) version (v) 5 =\< grade 1; subjects may be receiving equivalent to prednisone at a maximum dose of 20 mg/day orally
* Subjects must be age \>= 18 years
* Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%)
* Patients must have normal organ and marrow function, independent of transfusion or growth factor support within 14 days before enrollment; patients should not receive growth factors or transfusions for at least 7 days prior to the first dose of study drug, with the exception of pegylated GCSF (pegfilgrastim) and darbepoetin, which require at least 14 days prior to screening and enrollment
* Absolute neutrophil count (ANC) \>= 750 cells/uL (within 14 days before enrollment)
* Platelets \>= 50,000 cells/uL (within 14 days before enrollment)
* Total bilirubin =\< 1.5 times upper limit of normal unless due to Gilbert's disease (within 14 days before enrollment)
* Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 times upper limit of normal (within 14 days before enrollment)
* Calculated creatinine clearance \>= 30 mL/min by Cockcroft-Gault (within 14 days before enrollment)
* Corrected QT interval (QTc) =\< 480 ms (within 14 days before enrollment)
* Prothrombin time (PT)/international normalized ratio (INR) \< 1.5 times upper limit of normal (within 14 days before enrollment)
* Partial thromboplastin time (PTT) \< 1.5 times upper limit of normal (within 14 days before enrollment)
* The effects of ibrutinib and PD 0332991 (palbociclib) on the developing human fetus are unknown; for this reason and because tyrosine kinase inhibitors as well as other therapeutic agents used in this trial may be teratogenic, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation; female patients who are of non-reproductive potential include the following: post-menopausal by history - no menses for \>= 1 year; OR history of hysterectomy; OR history of bilateral tubal ligation; OR history of bilateral oophorectomy); female patients of childbearing potential must have a negative serum pregnancy test upon study entry; male and female patients who agree to use highly effective methods of birth control (e.g., condoms, implants, injectables, combined oral contraceptives, some intrauterine devices \[IUDs\], complete sexual abstinence, or sterilized partner) during the period of therapy and for 90 days after the last dose of study drug
* Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately; men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 4 months after completion of ibrutinib administration; subjects should be offered the opportunity to bank sperm or eggs prior to initiation of study drug
* Subjects with currently active, clinically significant hepatic impairment (\> moderate hepatic impairment according to the National Cancer Institute (NCI)/Child Pugh classification)
* Subjects requiring daily corticosteroids at a prednisone equivalent of \> 20 mg daily should not be enrolled; if corticosteroids can be discontinued (or reduced to \< 20 mg per day or prednisone equivalent), the discontinuation or dose reduction should be done at least 7 days prior to the first dose
* Subjects should be willing to undergo a research related biopsy prior to treatment and at the time of progression
* Subjects must give informed consent and must be willing and able to comply with the scheduled visits, treatment plans, laboratory tests, and other procedures

Exclusion Criteria:

* Subjects with known or suspected central nervous system (CNS) involvement are not eligible
* Subjects with serologic status reflecting active viral hepatitis B or C infection are not eligible; subjects that are positive for hepatitis B core antibody positive hepatitis B surface antigen (HBsAg), or hepatitis C antibody must have a negative polymerase chain reaction (PCR) prior to enrollment. (PCR-positive patients will be excluded.)
* Subjects with uncontrolled human immunodeficiency virus (HIV) are not eligible; controlled HIV is defined as a CD4 count \> institutional lower limit of normal and no current co-infection; uncontrolled HIV is all other HIV infection; note that patients with controlled infection should be allowed to participate only if they are not receiving prohibited cytochrome P450 (CYP) interactive medications
* Subjects unable to swallow capsules or with disease significantly affecting gastrointestinal function and/or inhibiting small intestine absorption, such as malabsorption syndrome, resection of the stomach or small bowel, partial or complete bowel obstruction, or symptomatic inflammatory bowel disease are not eligible
* Subjects with uncontrolled illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements are not eligible; recent infections requiring systemic treatment need to have completed therapy \> 14 days before the first dose of the study drugs
* Subjects with uncontrolled autoimmune hemolytic anemia or idiopathic thrombocytopenic purpura (ITP) resulting in declining platelet or hemoglobin levels within the 4 weeks prior to first dose of study drug are not eligible
* Patients with transfusion-dependent thrombocytopenia are not eligible
* Subjects with acute coronary syndrome within 6 months prior to enrollment or has New York Heart Association (NYHA) class III or IV heart failure, uncontrolled angina, severe uncontrolled ventricular arrhythmias, or electrocardiographic evidence of acute ischemia or active conduction system abnormalities are not eligible; prior to study entry, any electrocardiogram (ECG) abnormality at screening has to be documented by the investigator as not medically relevant
* Subjects with a history of stroke or intracranial hemorrhage within 6 months prior to enrollment are not eligible
* Subjects with a history of malignancy are not eligible with the exception of the following:

  * Malignancy treated within curative intent and with no evidence of active disease present for more than 3 years prior to screening and felt to be at low risk for recurrence by treating physician
  * Adequately treated non-melanomatous skin cancer or lentigo maligna melanoma without current evidence of disease
  * Adequately treated cervical carcinoma in situ without current evidence of disease
* Female subjects that are pregnant or breastfeeding are not eligible
* Subjects that have received anticoagulation therapy with warfarin or equivalent vitamin K antagonists within the last 28 days are not eligible
* Subjects with a bleeding diathesis (e.g., von Willebrand's disease or hemophilia) are not eligible
* Subjects that have undergone major surgery within 4 weeks prior to the first dose of study drug are not eligible
* Subjects receiving other investigational agents are not eligible
* Subjects who received a strong CYP3A inhibitor within 7 days prior to the first dose of study drug, or patients who require continuous treatment with a strong CYP3A inhibitor are not eligible
* Subjects receiving systemic immunosuppressant therapy (e.g., cyclosporine, tacrolimus, etc.) within 28 days of the first dose of study drug are not eligible
* Subjects that have been vaccinated with live, attenuated vaccines within 4 weeks of the first dose of study drug are not eligible