Acalabrutinib, Obinutuzumab, and Glofitamab for the Treatment of Relapsed or Refractory Mantle Cell Lymphoma

Inclusion Criteria: * Documented informed consent of the participant and/or legally authorized representative * Be willing to provide tissue from a fresh core or excisional biopsy (performed as standard of care) of a tumor lesion prior to starting study therapy or from diagnostic tumor biopsies * If unavailable, exceptions may be granted with study principal investigator (PI) approval * Age: \>= 18 years * Eastern Cooperative Oncology Group (ECOG) =\< 2 * Histologically confirmed diagnosis of mantle cell lymphoma according to the World Health Organization (WHO) classification * Relapsed or refractory disease after at least 1 prior line of systemic therapy * Relapse must have been confirmed histologically with hematopathology review at the participating institution. Exceptions may be granted with study PI approval * Tumor must be positive for CD20 by immunohistochemistry or flow cytometry after the most recent therapy * Active disease requiring treatment per treating physician's decision * Radiographically measurable disease by Lugano criteria (e.g., one or more nodal sites of disease \>= 1.5 cm and/or extranodal sites of disease \>= 1.0 cm in longest dimension) * If measurable bone marrow involvement or circulating disease has been confirmed in the absence of radiographically measurable disease, exceptions may be granted with study PI approval * Fully recovered from the acute toxic effects (except alopecia) to =\< grade 1 to prior anti-cancer therapy * Without bone marrow involvement: Absolute neutrophil count (ANC) \>= 1,000/mm\^3 With bone marrow involvement: No minimal requirement * NOTE: Growth factor is not permitted within 7 days prior to screening unless cytopenia is secondary to disease involvement. * Without bone marrow involvement: Platelets \>= 75,000/mm\^3 With bone marrow involvement: Platelets \>= 30,000/mm\^3 * NOTE: Platelet transfusions are not permitted within 7 days prior to screening unless cytopenia is secondary to disease involvement * Hemoglobin \>= 8 g/dL unless anemia is secondary to disease involvement * NOTE: Erythropoietin and/or red blood cell transfusions are not permitted within 7 days prior to screening. * Total bilirubin =\< 1.5 x upper limit of normal (ULN) (If hepatic involvement by lymphoma, or Gilbert's disease: =\< 3X ULN) * Aspartate aminotransferase (AST) =\< 2.5 x ULN (If hepatic involvement by lymphoma: AST =\< 5 x ULN) * Alanine aminotransferase (ALT) =\< 2.5 x ULN (If hepatic involvement by lymphoma: ALT =\< 5 x ULN) * Normal creatinine (Cr) level per local laboratory reference range or creatinine clearance of \>= 50mL/min per 24 hour urine test or the Cockcroft-Gault formula * If not receiving anticoagulants: International normalized ratio (INR) OR prothrombin (PT) =\< 1.5 x ULN If on anticoagulant therapy: PT must be within therapeutic range of intended use of anticoagulants * If not receiving anticoagulants: Activated partial thromboplastin time (aPTT) =\< 1.5 x ULN If on anticoagulant therapy: aPTT must be within therapeutic range of intended use of anticoagulants * Left ventricular ejection fraction (LVEF) \>= 40% * Oxygen (O2) saturation \>= 92% on room air * Women of childbearing potential (WOCBP): negative urine or serum pregnancy test * NOTE: If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required * Agreement by females of childbearing potential\* to abstain from heterosexual intercourse or use two adequate methods of birth control, including at least one method with a failure rate of \< 1% per year, during the treatment period until at least 7 days after the final dose of acalabrutinib; at least 2 months after the final dose glofitamab; at least 18 months after the dose of obinutuzumab; and at least 4 months after the last dose of tocilizumab (as applicable). Women must refrain from donating eggs during this same period Agreement by males to abstain from heterosexual intercourse or use a condom with female partners of childbearing potential or pregnant female partners during the treatment period and for at least 2 months after the last dose of glofitamab, and at least 4 months after the last dose of obinutuzumab, or tocilizumab (as applicable) to avoid exposing the embryo. Men must refrain from donating sperm during this same period * Childbearing potential defined as not being surgically sterilized (men and women) or have not been free from menses for \> 1 year (women only) with no identified cause other than menopause * Examples of contraceptive methods with a failure rate of \< 1% per year include bilateral tubal ligation, male sterilization, hormonal contraceptives that inhibit ovulation, hormone releasing intrauterine devices, and copper intrauterine devices Exclusion Criteria: * Prior treatment with a T-cell engaging bispecific antibody * Prior therapeutic intervention with any of the following: therapeutic anti-cancer antibodies within 4 weeks (i.e. rituximab); radio- or toxin-immunoconjugates within 4 weeks; all other chemotherapy or radiation therapy within 2 weeks prior to day 1 of protocol therapy * Prior exposure to a BTK inhibitor (including but not limited to ibrutinib, acalabrutinib, zanubrutinib, and pirobrutinib) for more than 180 cumulative days prior to enrollment. Patients with =\< 180 cumulative days on BTK inhibitor prior to enrollment are allowed, as long as they did not progress on treatment. * Prior chimeric antigen receptor (CAR) T cell therapy within 6 months of day 1 of protocol therapy * Prior allogeneic stem cell transplant * Autologous hematopoietic stem cell transplant within 3 months of day 1 of protocol therapy * Major surgical procedure (under general anesthesia) within 30 days of day 1 of protocol therapy. * Note: If a subject had major surgery, they must have recovered adequately from any toxicity and/or complications from the intervention before the first dose of study drug * Systemic steroid therapy for any cause must be tapered down to =\< 20 mg/day prednisone or equivalent. Exceptions are: * Use of brief ( =\< 7 days) course of high dose corticosteroids (100 mg/day prednisone or equivalent) prior to initiation of study therapy for control of lymphoma-related symptoms * Inhaled or topical steroids * Use of mineralocorticoids for management of orthostatic hypotension * Use of physiologic doses of corticosteroids for management of adrenal insufficiency * Prior treatment with systemic immunosuppressive medications (including, but not limited to, cyclophosphamide, azathioprine, methotrexate, thalidomide, and anti-tumor necrosis factor agents), within 2 weeks or five half-lives (whichever is shorter) prior to first dose of study treatment * Live virus vaccines within 30 days prior to day 1 of protocol therapy or planned administration of live virus vaccines during glofitamab therapy * Requires treatment with a strong cytochrome P450 3A4 (CYP3A4) inhibitor/inducer * History of solid organ transplantation * Known history of hypersensitivity or anaphylaxis to study drug(s) including active product or excipient components, including prior monoclonal antibody therapy * Concurrent participation in another therapeutic clinical trial * History of prior malignancy. Exceptions include malignancy treated with curative intent and no known active disease present for \>= 2 years prior to initiation of protocol therapy; adequately treated non-melanoma skin cancer or lentigo maligna (melanoma in situ) without evidence of disease; adequately treated in situ carcinomas (e.g., cervical, esophageal, etc.) without evidence of disease; asymptomatic prostate cancer managed with "watch and wait" strategy * Prior history of myeloid malignancies including myelodysplastic syndrome (MDS) or presence of cytogenetic and/or molecular abnormalities known to be associated with MDS or myeloproliferative neoplasms (MPN) (e.g. del 5q, chr 7 abn, JAK2 V617). Any evidence of clonal hematopoiesis in the screening bone marrow biopsy should be discussed with the study PI prior to enrollment. * Malabsorption syndrome, disease significantly affecting gastrointestinal function, or resection of the stomach or small bowel that is likely to affect absorption, symptomatic inflammatory bowel disease, partial or complete bowel obstruction, or gastric restrictions and bariatric surgery, such as gastric bypass * Known active central nervous system (CNS) involvement by lymphoma, including leptomeningeal involvement * Current or history of CNS disease, such as stroke, epilepsy, CNS vasculitis, or neurodegenerative disease. * Note: Subjects with a history of stroke who have not experienced a cerebrovascular accident (CVA), ischemic stroke or transient ischemic attack (TIA) within the past 2 years and have no residual neurologic deficits, as judged by the investigator, are allowed * Known history of progressive multifocal leukoencephalopathy (PML) * Known or suspected history of hemophagocytic lymphohistiocytosis (HLH) * Presence of a gastrointestinal ulcer diagnosed by endoscopy within 3 months before screening * Known bleeding disorders (e.g., von Willebrand's disease or hemophilia) * Clinically significant cardiovascular disease such as symptomatic arrhythmias, congestive heart failure, or myocardial infarction within 6 months of screening, any class III or IV cardiac disease as defined by the New York Heart Association (NYHA) Functional Classification, or any class C or D cardiac disease as defined by the NYHA Objective Assessment. * NOTE: Subjects with controlled, asymptomatic atrial fibrillation/flutter can enroll on study * Inability to swallow and retain an oral medication * Known active bacterial, viral, fungal, mycobacterial, parasitic, or other infection (excluding fungal infections of nail beds) at study enrollment or any major episode of infection (as evaluated by the investigator) within 4 weeks prior to day 1 of protocol therapy * Clinically significant history of liver disease, including viral or other hepatitis, or cirrhosis * Suspected or latent tuberculosis * Known active hepatitis B virus (HBV) or hepatitis C virus (HCV) infection. Testing to be done only in patients suspected of having infections or exposures. Patients with history of HBV infection (defined as negative hepatitis B surface antigen \[HBsAg\] and positive hepatitis B core antibody \[HBcAb\]) are eligible if HBV DNA is undetectable, if they are willing to undergo DNA testing on day 1 of every cycle and every three months for at least 12 months after the last cycle of study treatment and appropriate antiviral therapy. Patients who are positive for HCV antibody are eligible if polymerase chain reaction (PCR) is negative for HCV ribonucleic acid (RNA) (PCR testing only required if HCV antibody testing is positive). * Known active human immunodeficiency virus (HIV) infection. Subjects who have an undetectable or unquantifiable HIV viral load with CD4 \> 200 and are on highly active antiretroviral therapy (HAART) medication are allowed. Testing to be done only in patients suspected of having infections or exposures * Females only: Pregnant or breastfeeding, or intending to become pregnant, or unable/unwilling to comply with pregnancy testing and birth control measures * Any other condition that would, in the Investigator's judgment, contraindicate the patient's participation in the clinical study due to safety concerns with clinical study procedures * Prospective participants who, in the opinion of the investigator, may not be able to comply with all study procedures (including compliance issues related to feasibility/logistics)