  In this paper we present a novel CPT symmetry test in the neutral kaon system
based, for the first time, on the direct comparison of the probabilities of a
transition and its CPT reverse. The required interchange of "in"
$\leftrightarrow$ "out" states for a given process is obtained exploiting the
Einstein-Podolsky-Rosen correlations of neutral kaon pairs produced at a
$\phi$-factory. The observable quantities have been constructed by selecting
the two semileptonic decays for flavour tag, the $\pi\pi$ and $3\pi^0$ decays
for CP tag and the time orderings of the decay pairs. The interpretation in
terms of the standard Weisskopf-Wigner approach to this system, directly
connects CPT violation in these observables to the violating $\Re\delta$
parameter in the mass matrix of $K^0$-$\bar{K^0}$, a genuine CPT violating
effect independent of $\Delta \Gamma$ and not requiring the decay as an
essential ingredient. Possible spurious effects induced by CP violation in the
decay and/or a violation of the $\Delta S= \Delta Q$ rule have been shown to be
well under control. The proposed test is thus fully robust, and might shed
light on possible new CPT violating mechanisms, or further improve the
precision of the present experimental limits. It could be implemented at the
DA$\Phi$NE facility in Frascati, where the KLOE-2 experiment might reach a
statistical sensitivity of $\mathcal{O}(10^{-3})$ on the newly proposed
observable quantities.
