Nivolumab With and Without Ipilimumab and Radiation Therapy in Treating Patients With Recurrent or Resectable Undifferentiated Pleomorphic Sarcoma or Dedifferentiated Liposarcoma Before Surgery

Inclusion Criteria: * Adult subjects with treatment naive primary or locally recurrent dedifferentiated liposarcoma (DDLPS) of the retroperitoneum or undifferentiated pleomorphic sarcoma (UPS) of the trunk or extremity will be eligible for inclusion in this study only if all of the following criteria apply. * Patients must be capable of giving written informed consent, which includes compliance with the requirements and restrictions listed in the consent form. * Patients must have disease determined to be surgically resectable and candidates for upfront surgery as agreed upon by a multidisciplinary consensus (Surgical Oncology, Medical Oncology, Radiation Oncology) after presentation at sarcoma multidisciplinary conference. Resectable tumors are defined as having no significant vascular, neural or bony involvement. Only cases where a complete surgical resection can safely be achieved are defined as resectable. * Patients will be evaluated by the anesthesia team prior to surgery. * Patient must have recent imaging (computed tomography \[CT\] or magnetic resonance imaging \[MRI\], as appropriate) within 4 weeks of trial enrollment, demonstrating measurable disease as defined by RECIST 1.1. * Patients must have at least one tumor amenable to serial biopsy in clinic or be willing to undergo serial biopsies through image-guided procedures during the neoadjuvant phase of the protocol. Patients must be willing to provide tumor samples at the time points. * Patients must be medically fit to undergo surgery as determined by the treating medical and surgical oncology team and have Eastern Cooperative Oncology Group (ECOG) performance status 0-2. * Patients must have life expectancy \> 6 months. * Patients must be immunotherapy-naive. Those who have previously been treated with conventional chemotherapy for a prior history of sarcoma in the adjuvant setting may be included. * White blood cell count \> 3 K/uL. * Absolute neutrophil count (ANC) \> 1 K/uL. * Hemoglobin \> 9 g/dL. * Platelets \> 100 K/mm\^3. * Serum creatinine =\< 2 mg/dL OR creatinine clearance \> 50 mL/min. * Aspartic transaminase (AST) =\< 1.5 x upper limit of normal (ULN). * Alanine transaminase (ALT) =\< 1.5 x ULN. * Bilirubin =\< 1.5 x ULN. * Women are eligible to participate if: Non-childbearing potential defined as pre-menopausal females with a documented tubal ligation or hysterectomy; or postmenopausal defined as 12 months of spontaneous amenorrhea \[in questionable cases a blood sample with simultaneous follicle stimulating hormone (FSH) \> 40 MlU/mL and estradiol \< 40 pg/mL (\< 140 pmol/L) is confirmatory\]. Females on hormone replacement therapy (HRT) and whose menopausal status is in doubt will be required to use one of the contraception methods if they wish to continue their HRT during the study. Otherwise, they must discontinue HRT to allow confirmation of post-menopausal status prior to study enrollment. For most forms of HRT, at least 2-4 weeks will elapse between the cessation of therapy and the blood draw; this interval depends on the type and dosage of HRT. Following confirmation of their post-menopausal status, they can resume use of HRT during the study without use of a contraceptive method. * Childbearing potential and agrees to use method(s) of contraception. For a teratogenic study drug and/or when there is insufficient information to assess teratogenicity (preclinical studies have not been done), a highly effective method(s) of contraception (failure rate of less than 1% per year) is required. The individual methods of contraception and duration should be determined in consultation with the investigator. Women of childbearing potential (WOCBP) must follow instructions for birth control when the half-life of the investigational drug is greater than 24 hours, contraception should be continued for a period of 30 days plus the time required for the investigational drug to undergo five half-lives. WOCBP should use an adequate method to avoid pregnancy for 5 months (30 days plus the time required for nivolumab to undergo five half-lives) after the last dose of investigational drug. * WOCBP must have a negative serum or urine pregnancy test (minimum sensitivity 25 IU/L or equivalent units of human chorionic gonadotropin \[HCG\]) within 24 hours prior to the start of investigational product. * Men who are sexually active with WOCBP must use any contraceptive method with a failure rate of less than 1% per year. The investigator shall review contraception methods and the time period that contraception must be followed. Men who are sexually active with WOCBP must follow instructions for birth control when the half-life of the investigational drug is greater than 24 hours, contraception should be continued for a period of 90 days plus the time required for the investigational drug to undergo five half-lives. The half-life of nivolumab and ipilimumab is up to 25 days and 18 days, respectively. Therefore, men who are sexually active with WOCBP must continue contraception for 7 months (90 days plus the time required for nivolumab to undergo five half-lives) after the last dose of investigational drug. * Women who are not of childbearing potential (i.e., who are postmenopausal or surgically sterile) and azoospermic men do not require contraception. * Women must not be breastfeeding. * Other terms for undifferentiated pleomorphic sarcoma (UPS) may include, but are not limited to: pleomorphic undifferentiated sarcoma, unclassified spindle cell sarcoma, spindle cell sarcoma not otherwise specified, pleomorphic spindle cell sarcoma, pleomorphic fibroblastic sarcoma, undifferentiated high-grade pleomorphic sarcoma, pleomorphic sarcoma with prominent inflammation, pleomorphic sarcoma with giant cells, malignant fibrous histiocytoma (including storiform-pleomorphic and inflammatory subtypes), fibrosarcoma, and myxofibrosarcoma (at least intermediate grade; located deep to the fascia in muscle). Exclusion Criteria: * Disease that is considered surgically unresectable, including, but not limited to significant vascular, neural, or bone involvement, and in cases where a complete surgical resection cannot be safely performed. * Prior intra-abdominal surgery within 4 weeks of trial enrollment. * Prior chemotherapy or targeted small molecule therapy of the current sarcoma. In patients with locally recurrent disease, previous systemic chemotherapy of the primary tumor is allowed, as long as treatment was completed prior to study enrollment and patient has recovered (i.e., \< grade 1 or at baseline) from any adverse events due to previously administered agents. * Prior radiation therapy for sarcoma in the same area. * Active concurrent second malignancy. * Prior or concurrent immunotherapy, including treatment with an anti-PD-1, anti-PD-L1, anti-PD-L2, or anti-CTLA-4 antibody; tumor vaccines; interferon, or interleukins. * Prior malignancy active within the previous 2 years except for patient's prior diagnosis of sarcoma and locally curable cancers that have been apparently cured, such as basal or squamous cell skin cancer, superficial bladder cancer, or carcinoma in situ of the prostate, cervix, or breast with local control measures (surgery, radiation). * Non-oncology vaccine therapy used for prevention of infectious disease within 4 weeks of trial enrollment. * Pregnant or lactating female. * Unwillingness or inability to follow the procedures required in the protocol. * Current use of anticoagulants (warfarin, heparin, direct thrombin inhibitors) at therapeutic levels. * Any serious or uncontrolled medical disorder that, in the opinion of the investigator, may increase the risk associated with study participation or study drug administration, impair the ability of the subject to receive protocol therapy, or interfere with the interpretation of study results. * Subjects with active, known or suspected autoimmune disease. Subjects with vitiligo, type I diabetes mellitus, residual hypothyroidism due to autoimmune condition only requiring hormone replacement, psoriasis not requiring systemic treatment, or conditions not expected to recur in the absence of an external trigger are permitted to enroll. * Subjects with a condition requiring systemic treatment with either corticosteroids (\> 10 mg daily prednisone equivalents) or other immunosuppressive medications within 14 days of study drug administration. Inhaled or topical steroids and adrenal replacement doses \> 10 mg daily prednisone equivalents are permitted in the absence of active autoimmune disease. Brief dosing for contrast allergy prophylaxis is allowed. * Any positive test result for hepatitis B or C virus indicating acute or chronic infection. * Known history of testing positive for human immunodeficiency virus or known acquired immunodeficiency syndrome. * History of severe hypersensitivity reaction to any monoclonal antibody. * Subjects who are compulsorily detained for treatment of either a psychiatric or physical (infection disease) illness. * Prisoners or subjects who are involuntarily incarcerated.