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.