Inclusion Criteria:

* Baseline measurements and evaluations must be obtained within 6 weeks of randomization to the study. Abnormal PET or CT scans may constitute evaluable disease. Patient must have at least one objective measurable disease parameter. Measurable disease in the liver is required if the liver is the only site of lymphoma.
* MIPI score must be calculated and entered in Oncology Patient Enrollment Network (OPEN).
* Patients must have an Eastern Cooperative Oncology Group (ECOG) performance status score of 0-2.
* Patients must have untreated histologically confirmed mantle cell lymphoma, with cyclin D1 (BCL1) expression by immunohistochemical stains and/or t(11;14) by cytogenetics or fluorescent in situ hybridization (FISH). The diagnosis must be confirmed by formal hematopathology review at the enrolling center.
* Patients being treated with gastric reducing agents proton pump inhibitors must be switched to an alternative drug before starting acalabrutinib.
* Absolute neutrophil count (ANC) \>= 1,000/mcL (obtained with 14 days of randomization). If disease includes marrow involvement or hypersplenism, please reference the below revised ANC requirement:

* ANC \>= 500/mcL
* Platelets \>= 75,000 mcL (obtained with 14 days of randomization). If disease includes involvement or hypersplenism, please reference the below revised platelet requirement:

* Platelets \>= 25,000/mcL
* Total bilirubin =\< 2 x institutional upper limit of normal (ULN) (obtained with 14 days of randomization). If disease includes hepatic infiltration or is causing biliary obstruction, or if elevated bilirubin is due to Gilbert's disease, please reference the below revised bilirubin requirements:

* Bilirubin =\< 3 x institutional ULN
* Aspartate aminotransferase (AST) and alanine transaminase (ALT) =\< 2.5 x institutional ULN (obtained with 14 days of randomization). If disease includes hepatic infiltration or is causing biliary obstruction, or if elevated bilirubin is due to Gilbert's disease, please reference the below revised AST/ALT requirements:

* AST/ALT =\< 5 x institutional ULN
* Prothrombin time (PT)/international normalized ratio (INR) or partial thromboplastin time (aPTT) in the absence of lupus anticoagulant) \< 2 x institutional ULN (obtained with 14 days of randomization). Patients receiving anticoagulant therapy (other than warfarin or equivalent vitamin K antagonists which are excluded), higher INR/aPTT may be permitted to enroll to this study after discussion with the primary investigator (PI).
* Creatinine =\< institutional ULN, OR glomerular filtration rate (GFR) \>= 40 mL/min/1.73 m\^2 (obtained with 14 days of randomization).
* Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial.
* For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated.
* Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load.
* Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial.
* Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional classification. To be eligible for this trial, patients should be class 2B or better.
* Patients must have a QT interval (QTc) =\< 480 msec obtained within 14 days of randomization.
* Women must not be pregnant or breast-feeding due to the potential harm to an unborn fetus and possible risk for adverse events in nursing infants with the treatment regimens being used. Patients must also not expect to conceive or father children from the time of registration, while on study treatment, and until 12 months after the last dose of study treatment. All females of childbearing potential must have a blood test or urine study within 2 weeks prior to randomization to rule out pregnancy. A female of childbearing potential is any woman, regardless of sexual orientation or whether they have undergone tubal ligation, who meets the following criteria: has achieved menarche at some point, has not undergone a hysterectomy or bilateral oophorectomy; or has not been naturally postmenopausal (amenorrhea following cancer therapy does not rule out childbearing potential) for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months).
* Women of childbearing potential and sexually active males must agree to use accepted and effective method(s) of contraception or to abstain from sexual intercourse for the duration of their participation in the study and for 12 months after treatment ends.
* Patients are not eligible if they require treatment with a strong cytochrome P450 (CYP) 3A inhibitor.
* Patients may not have received the following within 7 days prior to the first dose of study drug:

* Strong and moderate CYP3A inhibitors
* Strong and moderate CYP3A inducers
* Patients are ineligible if they have any of the following:

* Malabsorption syndrome or disease significantly affecting gastrointestinal function.
* Active bleeding or history of bleeding diathesis (e.g. hemophilia or von Willebrand disease).
* Uncontrolled AIHA (autoimmune hemolytic anemia) or ITP (idiopathic thrombocytopenia purpura).
* Requires or receiving anticoagulation with warfarin or equivalent vitamin K antagonists (e.g. phenprocoumon) within 7 days of first dose of study drug.
* History of significant cerebrovascular disease/event, including stroke or intracranial hemorrhage, within 6 months before the first dose of study drug.
* Known active bacterial, viral, fungal, mycobacterial, parasitic, or other infections at study enrollment (defined as exhibiting ongoing signs/symptoms related to the infection and without improvement, despite appropriate antibiotics or other treatment).
* History of severe allergic reaction attributed to compounds of similar chemical or biologic composition to rituximab, bendamustine, cytarabine, or acalabrutinib.
* Patients must be able to fulfill one of the following eligibility requirements pertaining to biospecimen availability for submission following randomization:

* Archived formalin-fixed paraffin-embedded (FFPE) tumor tissue specimen from the original diagnostic biopsy is available for submission OR,
* If tumor tissue is not available, peripheral blood collected prior to initiation of protocol therapy will be submitted

* NOTE: Biospecimens must be submitted within 60 days following randomization to Adaptive Biotechnologies for ClonoSEQ ID molecular marker identification of unique clonal immunoglobulin deoxyribonucleic acid (DNA) sequence. If peripheral blood will be submitted, Adaptive Biotechnologies should be contacted prior to patient randomization for guidance pertaining to collection and submission requirements.