Autologous T Cells and Cyclophosphamide in Treating Patients With Soft Tissue Sarcoma That is Metastatic or Cannot Be Removed By Surgery

Inclusion Criteria: * Histopathological documentation of the diagnosis of synovial sarcoma or myxoid liposarcoma with metastatic or unresectable disease who have received an alkylating agent containing regimen (such as doxorubicin plus ifosphamide); this includes patients who received an alkylating agent as part of adjuvant therapy and then relapsed; patients who were treated on the PICCASSO trial who have progressed will be allowed on the study; "unresectable disease" shall include patients with locally advanced disease where a surgery could be attempted but where this surgery would be mutilating, debilitating and would likely fail to result in long-term disease free survival; in this setting a patient might reasonably choose to undergo salvage/second line systemic therapy but could also pursue aggressive surgical options as standard of care * Able to tolerate high-dose cyclophosphamide * NY-ESO-1 expression in \> 25% of tumor by immunohistochemistry (IHC) (at least 2+) * Expression of human leukocyte antigen (HLA)-A0201; high resolution HLA typing performed at any experienced HLA lab will be accepted * Zubrod performance status of '0-1' * Patients with metastatic disease must have bi-dimensionally measurable disease by palpation on clinical exam, or radiographic imaging (computed tomography \[CT\] scan) * All patients must have an electrocardiogram (ECG); all patients must have a normal stress test within 182 days prior to treatment * Patients must have already been leukapheresed on either protocol 1246 or 2365 prior to entry into this study; patients who are unable to have a leukapheresis product collected, for whatever reason, will be unable to participate in this study * If there is a patient with an NY-ESO-1 expressing sarcoma who would be otherwise eligible for the trial, where there has been disagreement between pathologists regarding the histopathologic diagnosis, eligibility will be decided on by the principal investigator (PI) * Patients must have had NY-ESO-1 specific cells already in production; patients must have NY-ESO-1 specific cells that have been generated and sorted; these cells may be either in the process of expansion or expanded and frozen at the time of enrollment * Patients with definitively treated brain metastasis and patients with 4 or fewer untreated lesions less than 1 cm each will be included at the discretion of the principal investigator (PI) * Patients must be off metformin at least 2 weeks before receiving T cell therapy * Patients must have hemoglobin A1C \< 8.5% Exclusion Criteria: * Patients for whom we are unable to generate NY-ESO-1 specific cells * Pregnant women, nursing mothers, men or 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 entry * Serum creatinine \> 1.5 mg/dL or glomerular filtration rate \< 50 * Significant hepatic dysfunction (serum glutamic oxaloacetic transaminase \[SGOT\] \> 150 IU or \> 3x upper limit of normal \[ULN\]) * Bilirubin \> 1.6 mg/dL * Prothrombin time (PT) \> 1.5 x control * Most patients with metastatic sarcoma will have pulmonary metastasis and it is expected that the majority will have some mild to moderate baseline shortness of breath; these patients will be allowed on study so long as their Eastern Cooperative Oncology Group (ECOG) performance status is 1; patients with severe pulmonary dysfunction (\>= grade 3 respiratory disorders as defined by Common Terminology Criteria for Adverse Events \[CTCAE\] version \[v\]4) will not allowed on study until their condition improves; however, patients who have recently experienced a decrease in their pulmonary function will be required to undergo pulmonary function testing; patients with a forced expiratory volume in one second (FEV1) \< 1.5L or diffusing capacity of carbon monoxide (DLco) (corrected \[corr\] for hemoglobin \[Hgb\]) \< 50% will be excluded; patients with a reversible cause of pulmonary dysfunction may undergo repeat testing and enroll if their pulmonary function tests (PFT's) meet criterion * All patients must have an echocardiogram (echo) showing ejection fraction (EF) \> 50% and normal troponin and creatine kinase (CK) MB (echo may be done at the time of stress test as a stress echo); furthermore the following significant cardiovascular abnormalities will be excluded: * Active, symptomatic congestive heart failure * Clinically significant hypotension * Symptoms of coronary artery disease * Presence of cardiac arrhythmias on EKG requiring drug therapy which has not been stable for at least 6 months * Patients with symptomatic untreated brain metastasis or asymptomatic untreated brain metastasis \> 1 cm will not be allowed to participate; additionally, patients with five or more untreated brain metastasis under 1 cm will not be allowed to participate; treatment may include surgery or stereotactic radiation at the discretion of the patient's treatment team; patients must be off steroids when starting therapy * Patients with active infections or oral temperature \> 38.2 C within 72 hours of study entry or systemic infection requiring chronic maintenance or suppressive therapy * Chemotherapeutic agents (standard or experimental or other immunosuppressive therapies) less than 3 weeks prior to T cell therapy; (patients with bulky disease may undergo cytoreductive chemotherapy but treatment will be discontinued at least 3 weeks prior to T cell infusion); patients may receive palliative radiation therapy two weeks prior to T cell infusion * Clinically significant autoimmune disorders or conditions of immunosuppression; patients with acquired immune deficiency syndrome (AIDS) or human immunodeficiency virus (HIV)-1 associated complex or known to be HIV antibody seropositive or known to be recently pathologic complete response (PCR)+ for hepatitis are not eligible for this study; virology testing will be done within 6 months of T cell infusion; the severely depressed immune system found in these infected patients and the possibility of premature death would compromise study objectives * Current treatment with steroids * Patients must not be receiving any other experimental drugs within 3 weeks of the initiation of treatment and must have recovered from all side effects of such therapy * Patients who were not negative for hepatitis B virus (HBV), hepatitis C virus (HCV) at the time of their leukapheresis on 1246 or 2365 must be retested to be sure they are PCR negative * Patients with a history of myocarditis, pericarditis, endocarditis