MC200708 Pemetrexed and Pembrolizumab for the Treatment of Recurrent and/or Metastatic Salivary Gland Cancer

Inclusion Criteria: * REGISTRATION - INCLUSION CRITERIA * Age \>= 18 years * Histologically confirmed diagnosis of recurrent or metastatic salivary gland cancer not amenable to curative-intent therapy * COHORT A1: Rochester Minnesota only: Diagnosis of adenoid cystic carcinoma (ACC) * Measurable disease as defined by Response Evaluation Criteria in Solid Tumors (RECIST) version (v)1.1 criteria * NOTE: Tumor lesions in a previously irradiated area are considered measurable disease if progression has been demonstrated in such lesions. Disease that is measurable by physical examination only is not eligible * Prior treatment: * Prior treatment with checkpoint inhibitor(s) allowed * Any number of lines of prior therapy in the recurrent/metastatic setting is permitted at the investigator's discretion * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0 or 1 * NOTE: PS must be assessed again within 7 days prior to first dose of study drug * Hemoglobin \>= 9.0 g/dL (obtained =\< 8 days prior to registration) * NOTE: Must be met without growth factor support and no transfusions \<14 days prior to testing * Absolute neutrophil count (ANC) \>= 1500/mm\^3 (obtained =\< 8 days prior to registration) * Platelet count \>= 100,000/mm\^3 (obtained =\< 8 days prior to registration) * Total bilirubin =\< 1.5 x upper limit of normal (ULN) (obtained =\< 8 days prior to registration) * Alanine aminotransferase (ALT) and aspartate transaminase (AST) =\< 2.5 x ULN (=\…