  In this paper we study the local theta correspondences between epipelagic
supercupsidal representations of a type I classical dual pair $(G,G')$ over
$p$-adic fields. We show that, besides an exceptional case, an epipelagic
supercupsidal representation $\pi$ of $\widetilde{G}$ lifts to an epipelagic
supercupsidal representation $\pi'$ of $\widetilde{G}'$ if and only if the
epipelagic data of $\pi$ and $\pi'$ are related by the moment maps.
