This phase I trial studies how well autologous NY-ESO-1-specific CD8-positive T lymphocytes (modified T lymphocytes [T cells]), chemotherapy, and aldesleukin with or without dendritic cell-targeting lentiviral vector ID-LV305 (LV305) and immunotherapeutic combination product CMB305 (CMB305) work in treating participants with sarcoma that has spread to other places in the body (advanced) or that has come back (recurrent). Modified T cells used in this study are taken from participants, are changed in a laboratory, and may "kill" some types of tumor cells. Drugs used in chemotherapy, such as cyclophosphamide, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Cyclophosphamide may help the body get ready to receive the modified T cells. Interleukins, such as aldesleukin, are proteins made by white blood cells and other cells in the body and may help regulate immune response. LV305 and CMB305 may help stimulate the immune system. Giving modified T cells, chemotherapy, aldesleukin, LV305, and CMB305 may work better in treating participants with sarcoma.

PRIMARY OBJECTIVES:

I. Evaluate the safety of adoptively transferred CD8 T cells targeting NY-ESO-1 positive (+) tumors given alone and in combination with antigen-specific vaccination.

II. Evaluate the functional and numeric in vivo persistence of NY-ESO-1-specific CD8 T-cells given alone and in combination with antigen-specific vaccination.

SECONDARY OBJECTIVES:

I. Evaluate the anti-tumor efficacy achieved following adoptive transfer of NY-ESO-specific CD8 T cells in combination with LV305 alone and with G305 vaccine in patients with advanced synovial and mixed round cell liposarcoma.

II. Evaluate the influence of antigen-specific vaccination on the induction of both CD8 and CD4 T cells to NY-ESO-1 and non-targeted tumor-associated antigens (antigen-spreading) and the correlation of these responses with clinical outcome.

OUTLINE: Participants are assigned to 1 of 3 groups.

COHORT 0: Participants receive cyclophosphamide intravenously (IV) over 30-60 minutes on day -2 and autologous NY-ESO-1-specific CD8-positive T lymphocytes IV over 60 minutes on day 0. Then, 6 hours later and twice a day for 14 days, receive aldesleukin subcutaneously (SC) in the absence of disease progression or unacceptable toxicity.

COHORT 1: Participants receive cyclophosphamide, autologous NY-ESO-1-specific CD8-positive T lymphocytes, and aldesleukin as in Cohort 0. Participants also receive dendritic cell-targeting lentiviral vector ID-LV305 intradermally (ID) on days 1, 22, 43, and 64 in the absence of disease progression or unacceptable toxicity.

After conclusion of study treatment, participants are followed up every 4 weeks for 168 days, then every 3 months for 24 months.

Inclusion Criteria:

• Histopathologic documentation of synovial sarcoma or myxoid liposarcoma with the diagnosis of advanced or recurrent disease who have received prior standard chemotherapy. Patients with other sarcoma subtypes if proven to be NY-ESO-1 positive and meeting all other eligibility criteria listed below will also be included.
• Tumor expression of NY-ESO-1 (2+ staining or > 25%) by immunohistochemistry (IHC).
• Expression of HLA-A*0201.
• Eastern Cooperative Oncology Group (ECOG)/ Zubrod performance status of '0-1'
• Life expectancy > 6 months.
• Electrocardiography (ECG) without evidence of clinically significant arrhythmia or ischemia.
• Women of childbearing potential (WOCBP) must be using at least one highly effective or two effective accepted methods of contraception to avoid conception throughout the study in such a manner that the risk of pregnancy is minimized. Suggested precautions should be used to minimize the risk or pregnancy for at least 1 month before start of therapy, and while women are on study for up to 3 months after T cell infusion and/or at least 3 months after the study agents LV305 or CMB305 are stopped. WOCBP include any female who has experienced menarche and who has not undergone successful surgical sterilization (hysterectomy, bilateral tubal ligation or bilateral oophorectomy) or is not postmenopausal.
• Men must be willing and able to use an acceptable method of birth control such as latex condom during the dosing period and for at least 3 months after completion of the study agent administration (T cell infusion and/or LV305 or CMB305) if their sexual partners are WOCBP.
• Willing and able to give informed consent.
• (Prior to treatment) Note: evaluate at least 1 week before T cell infusion. a. Adequate venous access - consider peripherally inserted central catheter (PICC) or central line. b. ECOG/Zubrod performance status of '0-1. c. Bi-dimensionally measurable disease by palpation on clinical exam, or radiographic imaging (X-ray, computed tomography [CT scan]). d. At least 4 weeks must have elapsed since the last chemotherapy, immunotherapy, radiotherapy or major surgery. At least 6 weeks for nitrosoureas, mitomycin C and liposomal doxorubicin. e. Toxicity related to prior therapy must either have returned to =< grade 1, baseline, or been deemed irreversible. f. Persons of reproductive potential must agree to use and utilize an adequate method of contraception throughout treatment and for at least 3 months after study drug is stopped. g. Willing and able to give informed consent.

Exclusion Criteria:

• Patients with active infections or oral temperature > 38.2 Celsius (C) within 72 hours of leukapheresis. The procedure may be deferred.
• Investigational therapy within 3 weeks.
• Prior administration of other NY-ESO-1 targeting immunotherapeutics.
• Significant immunosuppression from concurrent, recent (=< 4 weeks ago) or anticipated treatment with systemic corticosteroids at any dose, or other immunosuppressive medications such as methotrexate, cyclosporine, azathioprine (antihistamines, non-steroidal anti-inflammatory drugs and aspirin permitted) or conditions such as common variable hypogammaglobulinemia or exposures such as large field radiotherapy.
• Cancer therapies, including chemotherapy, radiation, biologic, or kinase inhibitors, granulocyte colony-stimulating factor (G-CSF), granulocyte-macrophage colony-stimulating factor (GM-CSF) within 3 weeks prior.
• Psychiatric, other medical illness or other condition that in the opinion of the principal investigator (PI) prevents compliance with study procedures or ability to provide valid informed consent.
• Significant autoimmune disease with the exception of alopecia, vitiligo, hypothyroidism or other conditions that have never been clinically active or were transient and have completely resolved and require no ongoing therapy.
• Myocardial infarction within 6 months of study initiation, active cardiac ischemia or New York Heart Association (NYHA) grade III or IV heart failure.
• Peripheral blood leukocyte count (white blood cells [WBC]) < 3000/mm^3.
• Absolute neutrophil count =< 1500/mm^3.
• Platelets < 75000/mm^3.
• Hemoglobin < 10 gm/dL.
• Alanine aminotransferase (ALT), and aspartate aminotransferase (AST) > 2.5 x upper limit of normal (ULN).
• Total serum bilirubin > 1.5 x ULN (patients with Gilbert's disease may be included if their total bilirubin is =< 3.0 mg/dL).
• Creatinine > 1.5 x ULN. If higher check 24hr clearance, if < 50 ml/min then patient will be excluded.
• INR (prothrombin time ratio) or partial thromboplastin time (PTT) > 1.5 x ULN (Please note: patients with hematopoietic cell transplantation (Hct) < 30%, WBC < 2500/mm/^3 and platelets < 50,000/mm^3 immediately prior to leukapheresis. The procedure may be deferred.)
• History of other cancer within 3 years (except non-melanoma cutaneous malignancies and cervical carcinoma in situ).
• Positive screening tests for human immunodeficiency virus (HIV), hepatitis (Hep) B, Hep C, active tuberculosis or recent (< 2 week ago) clinically significant infection or evidence of active HIV, Hep B, or Hep C. (Note: If positive results are not indicative of true active or chronic infection, the patient can be treated.)
• Brain metastases considered unstable as: a. without confirmed stability over 60 days in patients previously treated with prior surgery or radiation; OR b. associated with symptoms and/or findings; OR c. requiring corticosteroids or anticonvulsants in the prior 60 days.
• Pregnant, planning to become pregnant, or breast feeding.
• Known allergy(ies) to any component of CMB305 or LV305.
• Men or women of reproductive ability who are unwilling to use effective contraception and women of childbearing potential who are unwilling to undergo pregnancy testing before and during the study.
• Clinically significant pulmonary dysfunction, as determined by medical history and physical exam. Patients so identified will undergo pulmonary functions testing and those with forced expiratory volume in 1 second (FEV1) < 2.0 L or carbon monoxide diffusing capability (DLco) (correlation for hemoglobin [corr for Hgb]) < 75% will be excluded.
• Significant cardiovascular abnormalities as defined by any one of the following: a. congestive heart failure, b. clinically significant hypotension, c. symptoms of coronary artery disease, d. presence of cardiac arrhythmias on electrocardiography (EKG) requiring drug therapy, e. ejection fraction < 50 % (dobutimine stress echo).
• Active and untreated central nervous system (CNS) metastasis.
• Autoimmune disease: patients with a history of inflammatory bowel disease are excluded from this study, as are patients with a history of autoimmune disease (e.g. systemic lupus erythematosus, vasculitis, infiltrating lung disease) whose possible progression during treatment would be considered by the investigator to be unacceptable.
• Steroids are not permitted 3 days prior to T cell infusion and concurrently during therapy.
• No prisoners or children will be enrolled on this study.