In this paper, we present prospects for using the Jagiellonian positron emission tomograph (J-PET) detector to search for discrete symmetries violations in a purely leptonic system of the positronium atom. We discuss tests of CP and CPT symmetries by means of ortho-positronium decays into three photons. No zero expectation values for chosen correlations between ortho-positronium spin and momentum vectors of photons would imply the existence of physics phenomena beyond the standard model. Previous measurements resulted in violation amplitude parameters for CP and CPT symmetries consistent with zero, with an uncertainty of about 10−3. The J-PET detector allows to determine those values with better precision, thanks to the unique time and angular resolution combined with a high geometrical acceptance. Achieving the aforementioned is possible because of the application of polymer scintillators instead of crystals as detectors of annihilation quanta.