Eribulin Mesylate in Treating Patients With Recurrent or Metastatic Salivary Gland Cancer

Inclusion Criteria: * Patients must have histologically or cytologically documented salivary gland cancers; patients that do not have a salivary gland primary must have one of the following histologies - adenoid cystic carcinoma, mucoepidermoid carcinoma, acinic cell carcinoma * Patients must have recurrent and/or metastatic disease that is progressive and not amenable to surgery or curative radiotherapy occurring within 6 months of study entry, as evidenced by: at least a 20% increase in radiographically or clinically measurable disease, appearance of any new lesions, or deterioration in clinical status * Eastern Cooperative Oncology Group (ECOG) performance status of 0, 1, or 2 * Patients with measurable disease per RECIST 1.1 criteria * At least one lesion of \>= 1.5 cm in long-axis diameter for non lymph nodes or \>= 1.5 cm in short-axis diameter for lymph nodes which is serially measurable according to RECIST 1.1 using either computerized tomography (CT) or magnetic resonance imaging (MRI) * Lesions that have had radiotherapy must show evidence of progressive disease (PD) based on RECIST 1.1 to be deemed a target lesion * Absolute neutrophil count \>= 1,500/μL * Platelets \>= 100,000/μL * Creatinine clearance \>= 40 mL/min * Bilirubin =\< 1.5 upper limit of normal (ULN) * Alkaline phosphatase =\< 3 ULN; if total ALP is \> 3 x ULN (in the absence of liver metastasis) or \> 5 x ULN in subjects with liver metastasis AND the subject is known to have bone metastases, then liver ALP iso-enzyme should be used to assess liver function rather than total ALP * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 3 X ULN * Women of child-bearing potential (WOCP) and men must agree to use adequate contraception (hormonal or barrier method of birth control or abstinence) prior to study entry and for the duration of study participation * Life expectancy of \> 12 weeks * Signed and dated informed consent document indicating that the patient has been informed of all the pertinent aspects of the trial prior to enrollment Exclusion Criteria: * Patients with symptomatic central nervous system (CNS) metastases must have stable disease after treatment with surgery or radiation therapy * Second primary malignancy that is clinically detectable or clinically significant at the time of consideration for study enrollment * Radiotherapy within 14 days of study treatment * Major surgery within 21 days of study treatment; minor surgery within 2 weeks of study treatment; placement of vascular access device and biopsies allowed and is not considered major or minor surgery * Treatment with any chemotherapy or investigational agents within 4 weeks of the start of study treatment; subjects must have recovered from toxicities of prior therapy * Patients with peripheral neuropathy \>= grade 2 * Significant cardiovascular impairment: congestive heart failure \> class II according to the New York Heart Association (NYHA), unstable angina or myocardial infarction within 6 months of enrollment, or serious cardiac arrhythmia (\> grade 2) * Concomitant severe or uncontrolled medical disease * Significant psychiatric or neurologic disorder which would compromise participation in the study * Pregnant or breast-feeding females