  This work presents prospects for conducting a novel direct test of
time-reversal symmetry at the KLOE-2 experiment. Quantum entanglement of
neutral K meson pairs uniquely available at KLOE-2 allows to probe directly the
time-reversal symmetry (T) independently of CP violation. This is achieved by a
comparison of probabilities for a transition between flavour and CP-definite
states and its inverse obtained through exchange of initial and final states.
As such a test requires the reconstruction of the $K_L\to 3\pi^0$ decay
accompanied by $K_S\to\pi^{\pm}\ell^{\mp}\nu$ with good timing information, a
new reconstruction method for this process is also presented which is capable
of reconstructing the $K_L\to 3\pi^0$ decay with decay time resolution of
O(1$\tau_S$).
