Chemotherapy Before Surgery and Radiation Therapy or Surgery and Radiation Therapy Alone in Treating Patients With Nasal and Paranasal Sinus Cancer That Can Be Removed by Surgery

Inclusion Criteria:

* Patients must be \>= 18 years of age
* Patients must have an Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1
* Patients must have a general physical condition compatible with the proposed chemotherapy and surgery
* Patients must have stage T3 or T4a, histologically-confirmed NPNSCC requiring orbital or skull base resection: (Some patients with T4b disease deemed resectable by the treating surgeon can be included provided they fulfill all other eligibility criteria without involvement of the cavernous sinus
* Patients with T4b who have the following characteristics leading to a T4b definition but who in the opinion of the treating surgeon can have resectable disease can be included provided they fulfill all other eligibility criteria and provided they have one of the following presentations: 1. Invasion of orbital apex without involvement of the cavernous sinus 2. Dura invasion depending on extent of involvement and if total resection is deemed feasible. 3. Brain/middle cranial fossa invasion depending on extent of involvement if total resection is deemed feasible. 4. Nasopharynx invasion if very limited 5. Clivus invasion if very limited. (Surgeons are encouraged to consult with the protocol surgical chair for these particular cases). Stages T3 and T4a and selected T4b disease will be included regardless of nodal status (N0 or N1-3), provided that surgical therapy would require orbital or skull base resection. The surgical oncologist in each institution will determine the need for resection of the orbit OR base of skull at baseline for patients on both Arms A and B and following neoadjuvant chemotherapy for patients on Arm B using the score sheet provided at the end of the protocol.

  * Resection of skull base will be deemed necessary according to skull base bone erosion by computed tomography (CT) or marrow involvement by magnetic resonance imaging (MRI) is noted; for any disease abutting the skull base
  * Resection of orbital contents will be deemed necessary according to skull base society guidelines, based on involvement of periorbital fat documented by MRI imaging
* Patients must be deemed surgically resectable by the surgical teams at each institution and must have a determination of degree of anticipated structure preservation of orbit and skull base; this needs to be determined prior to randomization
* Patients must not have received previous irradiation for head and neck tumor, skull base, or brain tumors
* Absolute neutrophil count (ANC) \> 1500/mm\^3 (=\< 2 weeks prior to randomization)
* Hemoglobin (Hgb) \> 8.0 g/dL (=\< 2 weeks prior to randomization)
* Platelet count \> 100,000/mm\^3 (=\< 2 weeks prior to randomization)
* For patients receiving cisplatin, creatinine clearance must be \> 60 ml/min (=\< 2 weeks prior to randomization); creatinine clearance may be measured or calculated; if calculating, creatinine clearance, use the Cockcroft-Gault formula
* Total bilirubin within 1.5 x the upper limit of normal (ULN) (must be obtained =\< 2 weeks prior to randomization)
* Aspartate aminotransferase (AST) or alanine aminotransferase (ALT) must be within 1.5 x ULN allowing for eligibility (must be obtained \< 2 weeks prior to randomization)
* Alkaline phosphatase must be within 1.5 x ULN allowing for eligibility (must be obtained \< 2 weeks prior to randomization)
* Patients with a prior history of squamous cell or basal carcinoma of the skin or in situ cervical cancer must have been curatively treated
* Patient must not be pregnant or breast-feeding due to the potential harm to an unborn fetus and possible risk for adverse events in nursing infants with the chemotherapy and radiation being used; in addition, complications from pregnancy may interfere with the ability of patients to have an uninterrupted therapy

  * All patients of childbearing potential must have a blood test or urine study within 2 weeks prior to registration to rule out pregnancy
  * A patient of childbearing potential is anyone, regardless of sexual orientation or whether they have undergone tubal ligation, who meets the following criteria: 1) has achived menarche at some point, 2) has not undergone a hysterectomy or bilateral oophorectomy; or 3) has not been naturally postmenopausal for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months)
* Patient must not be expected conceive or father children by using an accepted and effective method(s) of contraception or by abstain from sexual intercourse for the duration of their participation in the study
* Patient must have measurable disease; MRI and/or PET/CT scans need to be performed within 2 weeks prior to randomization

Exclusion Criteria:

* Patients may not be receiving investigational agents at time of randomization, or at any time while on study and during the 4 weeks preceding randomization
* Patients with a history of allergic reactions attributed to compounds of similar chemical or biologic composition to docetaxel and/or both platinum-based chemotherapy agents are ineligible; patient must be able to receive at least one of the two proposed chemotherapy regimens
* Patients with evidence of distant metastases or leptomeningeal disease (LMD) are ineligbile
* Patients with uncontrolled inter-current illnesses which in the opinion of the investigator will interfere with the ability to undergo therapy including chemotherapy are ineligbile
* Patients with a history of a different malignancy are ineligible, unless the disease has not progressed for \>= 2 years
* Patient must not have current peripheral neuropathy \> grade 2 at time of randomization
* Patient must not have any co-existing condition that would preclude full compliance with the study; no prior history of severe hypersensitivity reaction to docetaxel or other drugs formulated with polysorbate 80