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 cau