Comparing the Effectiveness of the Immunotherapy Agents Rituximab or Mosunetuzumab in Patients With Nodular Lymphocyte-Predominant Hodgkin Lymphoma, NORM Trial

Inclusion Criteria: * Histopathologically confirmed diagnosis of NLPHL as confirmed by local pathologist's expert review. * Untreated NLPHL: stage IB to IV according to Cotswolds. The proportion of patients with stages I or II treated with consolidative radiotherapy will be capped at 40%. * Previously treated NLPHL, any stage. * According to the treating physician, the patient should not be observed and needs therapy, notably because of B-symptoms (unexplained fever \[temperature \> 38 degrees Celsius (\> 100.4 degrees Fahrenheit)\], weight loss \[unexplained loss of \> 10 percent of body weight over the past six months\], or drenching night sweats), symptomatic nodal or extranodal disease, or patient preferences. * Patients must have measurable disease according to the Lugano/Lymphoma Response to Immunomodulatory Therapy Criteria (LYRIC) classification. * Age \>= 18 years. Because no dosing or adverse event (AE) data are currently available on the use of mosunetuzumab in patients \< 18 years of age, children are excluded from this study. * Eastern Cooperative Oncology Group performance status =\< 2 (Karnofsky \>= 60%). * Absolute neutrophil count \>= 1,000/mcL. * Platelets \>= 100,000/mcL. * Total bilirubin =\< 1.5 institutional upper limit of normal (ULN), except in patients with Gilbert's syndrome as defined by \> 80% unconjugated bilirubin. * Aspartate aminotransferase (AST)(serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine transaminase (ALT)(serum glutami…