Testing Atezolizumab With Selinexor in People ≥ 12 Years Old With Alveolar Soft Part Sarcoma, AXIOM Trial

Inclusion Criteria:

* To be enrolled in the safety run-in, patients must have an advanced soft tissue sarcoma (not otherwise specified \[NOS\]). To be enrolled in the ASPS cohort, patients must have histologically or cytologically confirmed alveolar soft part sarcoma that is not curable by surgery

  * Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as ≥ 20 mm (≥ 2 cm) by chest x-ray or as ≥ 10 mm (≥ 1 cm) with CT scan, MRI, or calipers by clinical exam.
* Patients with unresectable, metastatic and measurable ASPS that is resistant/refractory to ICI treatment will be eligible for the ASPS cohort of this study if they show clinical and/or radiological evidence of disease progression after receiving prior ICI therapy (including history and increasing physical symptoms). On-study documentation will include a physician's rationale that supports evidence of clinical disease progression (i.e., increasing tumor pain)
* Age \>= 12 years. Because no dosing or adverse event data are currently available on the use of atezolizumab in combination with selinexor in patients ˂ 18 years of age, children \< 12 years of age are excluded from this study
* Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 70%)
* Absolute neutrophil count \>= 1,000/mcL
* Platelets \>= 100,000/mcL
* Hemoglobin \>= 9 g/dL
* Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN) (however, patients with known Gilbert disease who have serum bilirubin level =\< 3 x ULN may be enrolled)
* Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamic pyruvic transaminase \[SGPT\]) =\< 2.5 x institutional ULN or =\< 5 x ULN for patients with liver metastases
* Serum creatinine =\< 1.5 x institutional ULN OR creatinine clearance \>= 30 mL/min/1.73 m\^2 by Cockcroft-Gault
* Serum albumin \>= 2.5 g/dL
* Baseline sodium (Na+) \>= 130 mEq/L
* Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 28 days are eligible for this trial
* Administration of selinexor or atezolizumab may have an adverse effect on pregnancy and poses a risk to the human fetus, including embryo-lethality. Female patients of child-bearing potential and male patients must agree to use highly effective contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, for 90 days after the last dose of selinexor, and for 150 days after the last dose of atezolizumab, whichever is longer. Breastfeeding is not allowed while on selinexor or for 90 days after the last dose of selinexor. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately
* Ability to understand and the willingness to sign a written informed consent document
* Willingness to provide biopsy samples for research purposes, except patients enrolled on the safety run-in. Patients that cannot be safely biopsied may be considered for the study upon discussion with principal investigator
* Ability and willingness to swallow pills
* Vaccines intended to prevent severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) and coronavirus disease 2019 (COVID-19) are allowed

Exclusion Criteria:

* Malabsorption syndrome or other conditions that would interfere with intestinal absorption
* Patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study. However, the following therapies are allowed:

  * Hormone-replacement therapy or oral contraceptives
  * Herbal therapy \> 1 week prior to cycle 1, day 1 (herbal therapy intended as anti-cancer therapy must be discontinued at least 1 week prior to cycle 1, day 1)
  * Palliative radiotherapy for bone metastases \> 2 weeks prior to cycle 1, day 1
* Treatment with any other agent administered for the treatment of the patient's cancer, within five half-lives or 3 weeks prior to cycle 1, day 1, whichever is shorter
* History of malignancy other than ASPS prior to screening, with the exception of malignancies with a negligible risk of metastasis or death (e.g., 5-year overall survival \[OS\] rate \> 90%), such as adequately treated carcinoma in situ of the cervix, non-melanoma skin carcinoma, localized prostate cancer, ductal carcinoma in situ, or stage I uterine cancer
* Treatment with systemic immunostimulatory agents (including, but not limited to, interferon and interleukin 2 \[IL-2\]) within 4 weeks or 5 half-lives of the drug (whichever is longer) prior to initiation of study treatment
* Treatment with systemic immunosuppressive medications (including, but not limited to, prednisone, cyclophosphamide, azathioprine, methotrexate, thalidomide, and anti-tumor necrosis factor \[anti-TNF\] agents) within 2 weeks prior to cycle 1, day 1 or anticipation of need for systemic immunosuppressive medication during study treatment, with the following exceptions:

  * Patients who have received acute, low dose, systemic immunosuppressant medications or one-time pulse dose of systemic immunosuppressant medication (e.g., 48 hours of corticosteroids for a contrast allergy) are eligible after Principal Investigator confirmation has been obtained
  * Patients who have received mineralocorticoids (e.g., fludrocortisone), corticosteroids for chronic obstructive pulmonary disease (COPD) or asthma, or low-dose corticosteroids for orthostatic hypotension or adrenocortical insufficiency are eligible
* Patients taking bisphosphonate therapy for symptomatic hypercalcemia. Use of bisphosphonate therapy for other reasons (e.g., bone metastasis or osteoporosis) is allowed
* Patients with known primary central nervous system (CNS) malignancy or symptomatic CNS metastases are excluded, with the following exceptions:

  * Patients with asymptomatic untreated CNS disease may be enrolled, provided all of the following criteria are met:

    * Evaluable or measurable disease outside the CNS
    * No metastases to brain stem, midbrain, pons, medulla, or cerebellum
    * No history of intracranial hemorrhage or spinal cord hemorrhage
    * No ongoing requirement for dexamethasone for CNS; patients on a stable dose of anticonvulsants are permitted.
    * No neurosurgical resection or brain biopsy within 28 days prior to cycle 1, day 1
  * Patients with asymptomatic treated CNS metastases may be enrolled, provided all the criteria listed above are met as well as the following:

    * Radiographic demonstration of improvement upon the completion of CNS-directed therapy and no evidence of interim progression between the completion of CNS-directed therapy and radiographic screening for the current study
    * No stereotactic radiation or whole-brain radiation within 28 days prior to cycle 1, day 1
    * Screening CNS radiographic study \>= 4 weeks from completion of radiotherapy and \>= 2 weeks from discontinuation of corticosteroids
* History of severe allergic, anaphylactic, or other hypersensitivity reactions to chimeric antibodies, fusion proteins, or Chinese hamster ovary cell products or to any component of the atezolizumab formulation
* History of allergic reactions attributed to compounds of similar chemical or biologic composition to atezolizumab or selinexor
* Patients with uncontrolled intercurrent illness, prior malignancy, any other disease, metabolic dysfunction, physical examination finding, or clinical laboratory finding that contraindicates the use of an investigational drug, may affect the interpretation of the results, or may render the patient at high risk from treatment complications
* Patients who are pregnant or breastfeeding, or are expecting to conceive or father children within the projected duration of the study, starting with the screening visit through 5 months after the last dose of atezolizumab or 3 months after the last dose of selinexor, whichever is longer. A woman of childbearing potential (WOCBP) who has a positive urine pregnancy test (e.g., within 8 days) prior to treatment will be excluded from the study. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required. Women of childbearing potential must have a negative urine pregnancy test result within 8 days prior to initiation of study treatment. Pregnant women are excluded from this study because atezolizumab is a monoclonal antibody agent with the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with atezolizumab, breastfeeding should be discontinued if the mother is treated with atezolizumab. These potential risks may also apply to selinexor. Due to the potential risks, WOCBPs and men must agree to use highly effective contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and 5 months after completion of atezolizumab administration or 3 months after completion of selinexor administration, whichever is longer. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately
* Patients with prior allogeneic bone marrow transplantation or prior solid organ transplantation
* Known clinically significant liver disease, including active viral, alcoholic, or other hepatitis; cirrhosis; fatty liver; and inherited liver disease

  * Patients with past or resolved hepatitis B infection (defined as having a negative hepatitis B surface antigen \[HBsAg\] test and a positive anti-HBc \[antibody to hepatitis B core antigen\] antibody test) are eligible. For these patients, HBsAg and anti-HBc tests must be done within 28 days prior to enrollment
  * Patients positive for hepatitis C virus (HCV) antibody are eligible only if polymerase chain reaction (PCR) is negative for HCV ribonucleic acid (RNA). For these patients, an HCV RNA test must be done within 28 days prior to enrollment
* History or risk of autoimmune disease, including, but not limited to, myasthenia gravis, myositis, autoimmune hepatitis, systemic lupus erythematosus, rheumatoid arthritis, inflammatory bowel disease, antiphospholipid antibody syndrome, Wegener granulomatosis, Sjogren syndrome, Guillain-Barre syndrome, or multiple sclerosis, with the following exceptions:

  * Patients with a history of autoimmune-related hypothyroidism on a stable dose of thyroid replacement hormone may be eligible
  * Patients with controlled Type 1 diabetes mellitus on a stable insulin regimen may be eligible
  * Patients with eczema, psoriasis, lichen simplex chronicus, or vitiligo with dermatologic manifestations only (e.g., patients with psoriatic arthritis would be excluded) are permitted provided all of the following conditions are met:

    * Rash must cover less than 10% of body surface area (BSA)
    * Disease is well controlled at baseline and only requiring low potency topical steroids
    * No acute exacerbations of underlying condition within the last 12 months (not requiring psoralen plus ultraviolet A radiation \[PUVA\], methotrexate, retinoids, biologic agents, oral calcineurin inhibitors; high potency or oral steroids) within the previous 12 months
* History of idiopathic pulmonary fibrosis, pneumonitis (including drug induced), organizing pneumonia (i.e., bronchiolitis obliterans, cryptogenic organizing pneumonia, etc.), or evidence of active pneumonitis on screening chest computed tomography (CT) scan. History of radiation pneumonitis in the radiation field (fibrosis) is permitted
* Patients with significant cardiovascular disease (such as New York Heart Association class II or greater cardiac disease, myocardial infarction, or cerebrovascular accident) within 3 months prior to initiation of study treatment, unstable arrhythmia, or unstable angina are ineligible
* Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional classification. To be eligible for this trial, patients should be class IIB or better
* Patients with active tuberculosis (TB) are excluded
* Patients with mild or moderate signs or symptoms of infection within 2 weeks prior to cycle 1, day 1 (including, but not limited to, receiving oral or intravenous \[IV\] antibiotics) are excluded. Patients with severe infections within 4 weeks prior to cycle 1, day 1 (including, but not limited to, hospitalization for complications of infection, bacteremia, or severe pneumonia) are excluded. Patients receiving prophylactic antibiotics (e.g., for prevention of a urinary tract infection or chronic obstructive pulmonary disease) are eligible
* Major surgical procedure within 28 days prior to cycle 1, day 1
* Administration of a live, attenuated vaccine within 4 weeks before cycle 1, day 1 or anticipation that such a live, attenuated vaccine will be required during the study and up to 5 months after the last dose of atezolizumab

  * Influenza vaccination should be given during influenza season only (approximately October to March). Patients must not receive live, attenuated influenza vaccine within 4 weeks prior to cycle 1, day 1 or at any time during the study
* History of leptomeningeal disease
* Uncontrolled tumor-related pain. Patients requiring pain medication must be on a stable regimen at study entry, with no changes to their pain regimen in the 4 weeks prior to enrollment
* Symptomatic lesions (e.g., bone metastases or metastases causing nerve impingement) amenable to palliative radiotherapy should be treated prior to enrollment. Patients should be recovered from the effects of radiation. There is no required minimum recovery period
* Asymptomatic metastatic lesions that would likely cause functional deficits or intractable pain with further growth (e.g., epidural metastasis that is not currently associated with spinal cord compression) should be considered for loco-regional therapy if appropriate prior to enrollment
* Uncontrolled pleural effusion, pericardial effusion, or ascites requiring recurrent drainage procedures (once monthly or more frequently)
* Patients with indwelling catheters (e.g., PleurX \[registered trademark\]) are allowed
* Uncontrolled or symptomatic hypercalcemia (ionized calcium \> 1.5 mmol/L, calcium \> 12 mg/dL or corrected serum calcium \> ULN)
* Any other disease, metabolic dysfunction, physical examination finding, or clinical laboratory finding that contraindicates the use of an investigational drug, may affect the interpretation of the results, or may render the patient at high risk from treatment complications
* Current treatment with anti-viral therapy for HBV
* Treatment with investigational therapy within five half-lives or 28 days prior to initiation of study treatment, whichever is shorter