NY-ESO-1 Specific T Cells After Cyclophosphamide in Treating Patients With Advanced Synovial Sarcoma or Myxoid/Round Cell Liposarcoma

Inclusion Criteria: INCLUSION CRITERIA FOR SCREENING: * Medical history including histopathological documentation of sarcoma * Informed consent and Health Insurance Portability and Accountability Act (HIPAA) signing CRITERIA FOR LEUKAPHERESIS: * NY-ESO-1 expression in tumor by immunohistochemistry (IHC) is not required prior to screening consent; however, patients must have NY-ESO-1 expression to proceed with the leukapheresis; additionally patients must also meet the following criteria to proceed with leukapheresis (any exceptions to this will require prior approval by the Apheresis director and Principal Investigator): * Physical exam and Karnofsky performance status * Laboratory evaluation: * Complete blood count (CBC), differential and platelet count * Basic metabolic and hepatic function panels * Puget Sound Blood Center Recipient Donor Battery Panel * Pregnancy test for females of child-bearing potential * Pulse \> 45 and \< 120 * Weight \>= 45 kg * Temperature =\< 38 Celsius (C) (=\< 100.4 Fahrenheit \[F\]) * White blood cell (WBC) \>= 2,000 * Hematocrit (HCT) \>= 30% * Platelets \>= 75,000 * Research blood including 40 mL of blood in a heparinized tube for peripheral blood mononuclear cell (PBMC) collection and 10 mL of blood for serum collection (generally in a red top tube) within 30 days of leukapheresis collection INCLUSION CRITERIA FOR TREATMENT: * A diagnosis of synovial sarcoma or myxoid/round cell liposarcoma * Patients must have "advanced disease"; usually, this will mean metastatic disease; this may also be multiply recurrent disease or locally advanced disease; locally advanced disease is defined for this study as disease where a mutilating surgery is required and the patient is more likely than not to die of their disease despite an aggressive operation; patients with metastatic disease that has been resected or radiated are allowed to participate; Response Evaluation Criteria in Solid Tumors (RECIST) evaluable disease is not necessary for participation * NY-ESO-1 positive by IHC (for this study, even a small level of positivity is acceptable); for patients with \< 5% NY-ESO-1 by IHC positivity, the level of staining should be discussed with the patient and they should be informed that this will likely effect the efficacy of the therapy; this conversation must be documented in the patient's medical chart * Human leukocyte antigen (HLA)-A0201 or HLA-A2402 * Zubrod performance status of '0-1' * All patients must have an electrocardiogram (ECG) within 2 weeks of starting conditioning * All patients must have a stress test within 6 months of starting treatment showing no evidence of cardiac ischemia * All patients must have an echo or multi gated acquisition scan (MUGA) scan showing ejection fraction (EF) \> 50% and normal troponin and creatine kinase (CK) myoglobin binding (MB) (echo may be done at the time of stress test as a stress echo); within 6 months of starting treatment; however if the patient has received cardiotoxic chemotherapy such as Adriamycin, they must have had an echo or MUGA scan since completing this treatment * If there is a patient with an NY-ESO-1 expressing soft tissue sarcoma who would be otherwise eligible for the trial, which is either not synovial sarcoma (SS) or myxoid/round cell liposarcoma (MRCL) or there is controversy about the diagnosis, eligibility will be decided by the PI * Patients must have NY-ESO-1 specific cells already produced or in production; these cells may be either in the process of their final expansion (for fresh infusion) or expanded and frozen at the time of enrollment Exclusion Criteria: * Patients for whom we are unable to generate NY-ESO-1 specific cells * Pregnant women, nursing women, and men and women of reproductive ability who are unwilling to use effective contraception or abstinence; women of childbearing potential must have a negative pregnancy test within two weeks prior to study entry * Serum creatinine \> 1.6 mg/dL or glomerular filtration rate \< 50 * Serum glutamic oxaloacetic transaminase (SGOT) \> 150 IU or \> 3 x upper limit of normal * Bilirubin \> 1.6 mg/dL * Prothrombin time \> 1.5 x control * Active symptomatic congestive heart failure * Clinically significant hypotension (systolic blood pressure \[SBP\] \< 80 mm HG or symptomatic) * Newly diagnosed cardiac arrhythmia; patients with an arrhythmia that has been stable for at least 6 months will be allowed to participate * Known untreated central nervous system (CNS) metastasis; patients with CNS metastasis will be allowed on study once treated * Patients with active systemic infections requiring antibiotics or chronic maintenance/suppressive therapy; once the infection in question has resolved and patient is off antimicrobial treatment, patients may participate * Systemic anticancer treatments (including chemotherapy and biologics) less than 3 weeks prior to T cell therapy; locally directed therapy (e.g. radiation) 2 weeks prior to cell infusion * Known clinically significant autoimmune disorders requiring systemic immunosuppression for control * Patients who are known to be human immunodeficiency virus (HIV) positive are not eligible for this study * Current treatment with steroids * Known infection with hepatitis B virus (HBV) and hepatitis C virus (HCV); (if a patient was not tested at the time of their leukapheresis, they must be tested prior to receiving T cell infusion; if testing was done for leukapheresis, this is adequate and does not need to be repeated) * Patients with a known history of proven myocarditis, pericarditis, and/ or endocarditis