A Study of Mirdametinib in Combination With Palbociclib in People With Liposarcoma

Inclusion Criteria: Phase I only: * A diagnosis of unresectable, recurrent, or metastatic DDLPS * Measurable disease as defined by RECIST 1.1 Phase II only: * A diagnosis of unresectable, recurrent (e.g. recurrent retroperitoneal) or metastatic DDLPS * Any number of prior lines of therapy * Measurable disease and evidence of progression of disease as defined by RECIST 1.1 (including newly diagnosed disease, new disease sites in a patient who was previously NED, or a 20% growth of existing lesions within 6 months of registration) * Age ≥ 18 years * ECOG performance status ≤ 2 * Adequate organ and marrow function as defined below (ULN indicates institutional upper limit of normal): * Absolute neutrophil count ≥ 1.5 x 109/L * Hemoglobin ≥ 9.0 g/dL * Platelets ≥ 100 x 109/L * Total bilirubin ≤ 1.5 X ULN OR Direct bilirubin ≤ ULN for subjects with total bilirubin levels \> 1.5 ULN, except patients with Gilbert's disease (≤3x ULN) * AST (SGOT) /ALT (SGPT) ≤ 1.5 x institutional ULN * Creatinine Clearance ≥ 60 mL/min (calculated by Cockcroft-Gault method) * Adequate coagulation function, as determined by: * International Normalized Ratio (INR) ≤ 1.5 × ULN (Grade ≤ 1). If the participant receives anticoagulant therapy, the INR \> 1.5 × ULN is permitted but the dose must be stable for at least 2 weeks before the start of the study treatments. * PTT ≤ 1.5 × ULN. * Adequate cardiac function, as determined by: * Systolic blood pressure \< 160 mmHg and diastolic blood pressure \< 100 mmHg (Grade ≤ 2). * LVEF ≥ 50% by MUGA or ECHO. * No clinically significant ECG waveform abnormalities assessments at screening. * Adequate glycemic control, as determined by: * Fasting blood glucose level \< 125 mg/dL, or * Random blood glucose level \< 200 mg/dL. * Have normal serum calcium and phosphate levels (calcium level may be corrected for albumin level). * Have intraocular pressure ≤ 21 mmHg in both eyes * Women of child-bearing potential must agree to use highly effective contraceptive methods (hormonal or barrier method of birth control or abstinence) during the trial period through at least six months after the last dose. Male patients or their partners must be surgically sterile or agree to use adequate contraception while receiving trial treatment and for three months thereafter. Acceptable methods of contraceptive use by men or women are detailed in Section 15.3. * Ability to understand and the willingness to sign a written informed consent document. * Ability to swallow tablets or capsules * Patients with brain metastasis that have been treated with definitive surgery or radiation, and have been clinically stable for 3 months are eligible. Exclusion Criteria: * Patients who have not recovered from clinically significant adverse events of prior therapy to ≤ NCI CTCAE v5 Grade 1, except alopecia and stable neuropathy, which must have resolved to ≤ Grade 2 or baseline. * Patients receiving any other investigational agents. * Phase II only: Receipt of prior treatment with a selective CDK4 inhibitor or MEK inhibitor * Uncontrolled intercurrent illness including, but not limited to, known ongoing or active infection, including uncontrolled HIV, active hepatitis B or C, symptomatic congestive heart failure, unstable angina pectoris, uncontrolled cardiac arrhythmias, psychiatric illness/social situations that would limit compliance with study requirements, clinically significant interstitial lung disease or active noninfectious pneumonitis, or active infection requiring systemic therapy. * Patients with a CD4+ count of \> 300 and an undetectable viral load who are currently on HAART are eligible for inclusion. * Patients with NYHA class III or IV congestive heart failure within 6 months of study treatment will be excluded. * Patients with history of clinically significant cardiac disease (New York Heart Association Class III or IV), myocardial infarction, severe/unstable angina, coronary/peripheral artery bypass graft, symptomatic congestive heart failure, cerebrovascular accident, clinically significant transient ischemic attack, symptomatic pulmonary embolism, unexplained syncope, or long QT syndrome within 6 months before the start of study treatment will be excluded. * Pregnant women and women who are breast-feeding. * Prolonged QTcF \> 470ms at Screening, irrespective of sex. o If a single 12-lead electrocardiogram (ECG) or, for patients with prolonged QT intervals or other cardiac indications, a triplicate ECG should be performed. * Current Chronic Kidney Disease stage \> 3 or Creatinine Clearance \< 60 mL/min (calculated by Cockcroft-Gault method) * Current or history of Interstitial Lung Disease * History or current evidence of glaucoma or clinically significant abnormalities on the ophthalmological exam, including but not limited to cataract limiting the ability to examine the retina or any ophthalmological finding that could be a significant risk factor for RVO, retinopathy or neovascular macular degeneration. * Concurrent neuromuscular disorder that is associated with the potential of elevated CPK (e.g., inflammatory myopathies, muscular dystrophy, amyotrophic lateral sclerosis, spinal muscular atrophy). * Radiation therapy within 2 weeks prior to study Day 1 * Major surgery, other than diagnostic surgery, within 2 weeks prior to Cycle 1 Day 1, without complete recovery. * Patient is receiving systemic (oral or IV/SC) or ocular glucocorticoid therapy (with the exception of participants with endocrine deficiencies who are allowed to receive physiologic or stress doses of steroids, if necessary) within 14 days prior to first dose of study treatment * Known prior severe hypersensitivity to investigational product or any component in its formulation. o This includes hypersensitivity to imidazoles, such as clotrimazole, ketoconazole, miconazole and others in this drug class. Subjects with hypersensitivity to these agents will be excluded from enrollment. * History of significant toxicity related to prior CDK4/6, MEK, or ERK inhibitor requiring discontinuation of treatments with these agents. * Concurrent, clinically significant, active malignancies within 12 months of study enrollment * Current evidence of a disorder that could reduce the ability to swallow oral dosage forms or alter absorption of orally administered drugs. * Patients who require concomitant use of medications that strongly induce or inhibit CYP3A or UDP-glucuronosyltransferase (UGT) * Non-tolerable Grade 2 or ≥ Grade 3 neuropathy or evidence of unstable neurological symptoms within 4 weeks of Cycle 1 Day 1. Non-tolerable Grade 2 toxicities are defined as those with moderate symptoms that the subject is not able to endure for the conduct of instrumental activities of daily life or that persists ≥ 7 days.