This phase II trial compares three chemotherapy regimens consisting of bendamustine, rituximab, high dose cytarabine, and acalabrutinib and studies how well they work in treating patients with newly diagnosed mantle cell lymphoma. Drugs used in chemotherapy, such as bendamustine and cytarabine, work in different ways to stop the growth of cancer cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Monoclonal antibodies, such as rituximab, may interfere with the ability of cancer cells to grow and spread. Acalabrutinib may stop the growth of cancer cells by blocking some of the enzymes needed for cell growth. This study is being done to find out if one the drug combinations of bendamustine, rituximab, high dose cytarabine, and acalabrutinib is better or worse than the usual approach for mantle cell lymphoma.