  A paper of Reeder-Yu gives a construction of epipelagic supercuspidal
representations of $p$-adic groups. The input for this construction is a pair
$(\lambda, \chi)$ where $\lambda$ is a stable vector in a certain
representation coming from a Moy-Prasad filtration, and $\chi$ is a character
of the additive group of the residue field. We say two such pairs are
equivalent if the resulting supercuspidal representations are isomorphic. In
this paper we describe the equivalence classes of such pairs. As an
application, we give a classification of the simple supercuspidal
representations for split adjoint groups. Finally, under an assumption about
unramified base change, we describe properties of the Langlands parameters
associated to these simple supercuspidals, showing that they have trivial
L-functions and minimal Swan conductors, and showing that each of these simple
supercuspidals lies in a singleton L-packet.
