Search for possible violation of combined charge, parity, and time-reversal symmetries is yet another approach for a test of New Physics. Positronium being a bound state of electron and positron, is the lightest matter-antimatter system and at the same time eingenstate of the C and P operators, which makes it a unique probe in such endeavour. The experimental method consists of the measurement of angular correlations constructed based on the momenta and/or polarization of photons originating from the positronium annihilation, and the positronium spin orientation. TheJ-PET detector is the only device which enables determination of polarization of photons from positronium annihilation together with estimation of positronium spin axis on the event-by-event basis. This allows exploration of a new class of discrete symmetry-odd operators that has not been investigated before. With first measurements demonstrating such capabilities, we are able to reach the precision of CP and CPT tests at permill level. In this contribution we describe experimental techniques and new results of discrete symmetry tests in the decays of positronium