Inclusion Criteria:

          -  Documented informed consent of the participant and/or legally authorized
             representative

               -  Assent, when appropriate, will be obtained per institutional guideline

          -  Agreement to allow the use of archival tissue from diagnostic tumor biopsies

               -  If unavailable, exceptions may be granted with study principal investigator (PI)
                  approval

          -  Age: >= 18 years

          -  Eastern Cooperative Oncology Group (ECOG) =< 2

          -  Histologically confirmed mycosis fungoides (MF) or Sezary syndrome (SS). Safety
             lead-in: >= stage IIB OR >= stage IB-IIA folliculotropic/transformed MF. Phase 2: >=
             stage IB

               -  Stage of disease according to Tumor-Node-Metastasis-Blood (TNMB) classification

               -  Pathology report must be diagnostic or be consistent with MF/SS criteria

               -  SS is defined as meeting T4 plus B2 criteria; where the biopsy of erythrodermic
                  skin may only reveal suggestive but not diagnostic histopathologic features, the
                  diagnosis may be based on either node biopsy or fulfillment of B2 criteria

               -  For MF where the histological diagnosis by light microscopic examination is not
                  confirmed, diagnostic criteria that been recommended by the International Society
                  of Cutaneous Lymphomas (ISCL) should be used.

          -  Measurable disease per Modified Severity Weighted Assessment Tool (mSWAT) and/or
             Sezary count

          -  Baseline skin biopsy taken within 6 months available for central review submission

          -  Without bone marrow involvement: Absolute neutrophil count (ANC) >= 1,500/mm^3
             (performed within 7 days prior to day 1 of protocol therapy unless otherwise stated)

               -  NOTE: Growth factor is not permitted within 14 days of ANC assessment unless
                  cytopenia is secondary to disease involvement

          -  With bone marrow involvement: ANC >= 1,000/mm^3 (performed within 7 days prior to day
             1 of protocol therapy unless otherwise stated)

               -  NOTE: Growth factor is not permitted within 14 days of ANC assessment unless
                  cytopenia is secondary to disease involvement

          -  Without bone marrow involvement: Platelets >= 100,000/mm^3 (performed within 7 days
             prior to day 1 of protocol therapy unless otherwise stated)

               -  NOTE: Platelet transfusions are not permitted within 14 days of platelet
                  assessment unless cytopenia is secondary to disease involvement

          -  With bone marrow involvement: Platelets >= 75,000/mm3 (performed within 7 days prior
             to day 1 of protocol therapy unless otherwise stated)

               -  NOTE: Platelet transfusions are not permitted within 14 days of platelet
                  assessment unless cytopenia is secondary to disease involvement

          -  Total bilirubin =< 1.5 x upper limit of normal (ULN) (unless has Gilbert's
             disease)(performed within 7 days prior to day 1 of protocol therapy unless otherwise
             stated)

          -  Aspartate aminotransferase (AST) =< 2.5 x ULN (unless has Gilbert's disease)(performed
             within 7 days prior to day 1 of protocol therapy unless otherwise stated)

          -  Alanine aminotransferase (ALT) =< 2.5 x ULN (unless has Gilbert's disease)(performed
             within 7 days prior to day 1 of protocol therapy unless otherwise stated)

          -  Creatinine clearance of >= 60 mL/min per 24 hour urine test or the Cockcroft-Gault
             formula (unless has Gilbert's disease) (performed within 7 days prior to day 1 of
             protocol therapy unless otherwise stated)

          -  If not receiving anticoagulants: International normalized ratio (INR) OR prothrombin
             (PT) =< 1.5 x ULN (performed within 7 days prior to day 1 of protocol therapy unless
             otherwise stated)

          -  If on anticoagulant therapy: PT must be within therapeutic range of intended use of
             anticoagulants (performed within 7 days prior to day 1 of protocol therapy unless
             otherwise stated)

          -  If not receiving anticoagulants: Activated partial thromboplastin time (aPTT) =< 1.5 x
             ULN (performed within 7 days prior to day 1 of protocol therapy unless otherwise
             stated)

          -  If on anticoagulant therapy: aPTT must be within therapeutic range of intended use of
             anticoagulants (performed within 7 days prior to day 1 of protocol therapy unless
             otherwise stated)

          -  Hepatitis C virus (HCV)*, active hepatitis B virus (HBV) (surface antigen negative),
             and syphilis (rapid plasma reagin [RPR])

               -  If positive, hepatitis C ribonucleic acid (RNA) quantitation must be performed

          -  Meets other institutional and federal requirements for infectious disease titer
             requirements

               -  Note Infectious disease testing to be performed within 28 days prior to day 1 of
                  protocol therapy

          -  Subjects with MF and a history of staphylococcus colonization are eligible provided
             they continue to receive stable doses of prophylactic antibiotics

          -  Women of childbearing potential (WOCBP): negative urine or serum pregnancy test If the
             urine test is positive or cannot be confirmed as negative, a serum pregnancy test will
             be required

          -  Agreement by females and males of childbearing potential* to use an effective method
             of birth control or abstain from heterosexual activity for the course of the study
             through at least 3 months after the last dose of protocol therapy

               -  Childbearing potential defined as not being surgically sterilized (men and women)
                  or have not been free from menses for > 1 year (women only)

        Exclusion Criteria:

          -  Prior mogamulizumab

          -  Any systemic therapy, including monoclonal antibody within 28 days or 5 half-lives
             (whichever is shorter) of initiating protocol therapy

          -  Chemotherapy, radiation therapy, biological therapy, immunotherapy within 21 days
             prior to day 1 of protocol therapy

          -  Any skin-directed therapy within 14 days prior to initiating protocol therapy

          -  Any radiation therapy within 21 days prior to initiating protocol therapy

          -  Immunosuppressive medication within 14 days prior to the first dose of study
             treatment. The following are exceptions to this criterion:

               -  Intranasal, inhaled, topical or local steroid injections (e.g., intra-articular
                  injection) and are on stable dose for at least 28 days

               -  Systemic corticosteroids at physiologic doses of < 10 mg/day of prednisone or
                  equivalent

          -  Live, attenuated vaccine within 30 days prior to the first dose protocol therapy

          -  Disease free of prior malignancies for >= 5 years with the exception of:

               -  Currently treated squamous cell and basal cell carcinoma of the skin, or

               -  Carcinoma in situ of the cervix, or

               -  Surgically removed melanoma in situ of the skin (stage 0) with histological
                  confirmed free margins of excision, or

               -  Prostate cancer (T1a or T1b using the TNM [tumor, nodes, metastasis] clinical
                  staging system) that has/have been surgically cured, or

               -  Any other malignancy that has/have been curatively treated with surgery and/or
                  localized radiation

          -  Active infection requiring antibiotics

          -  Known hepatitis B or hepatitis C infection

          -  Other active malignancy

          -  Females only: Pregnant or breastfeeding

          -  Prior stem cell transplantation

          -  Acute infection requiring systemic treatment

          -  Conditions requiring chronic steroid or immunosuppressive treatment that likely need
             additional steroid or immunosuppressive treatments in addition to the protocol therapy

          -  Renal failure requiring hemodialysis or peritoneal dialysis

          -  Unstable cardiac disease as defined by one of the following:

               -  Cardiac events such as myocardial infarction (MI) within the past 6 months

               -  NYHA (New York Heart Association) heart failure class III-IV

               -  Uncontrolled atrial fibrillation or hypertension

          -  Major surgery (as defined by the investigator) within the 28 days prior to the first
             dose of study treatment

          -  Active or prior documented autoimmune or inflammatory disorders requiring therapy
             within the past 3 years prior to the start of treatment. The following are exceptions
             to this criterion:

               -  Vitiligo or alopecia

               -  Hypothyroidism (e.g., following Hashimoto syndrome) stable on hormone
                  replacement; or

               -  Psoriasis not requiring systemic treatment

          -  History of primary immunodeficiency

          -  Any other condition that would, in the Investigator's judgment, contraindicate the
             patient's participation in the clinical study due to safety concerns with clinical
             study procedures.

          -  Prospective participants who, in the opinion of the investigator, may not be able to
             comply with all study procedures (including compliance issues related to
             feasibility/logistics)