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