  Let $G$ be a connected reductive group over a finite field $\mathfrak{f}$ of
order $q$. When $q$ is small, we make further assumptions on $G$. Then we
determine precisely when $G(\mathfrak{f})$ admits irreducible, cuspidal
representations that are self-dual, of Deligne-Lusztig type, or both. Finally,
we outline some consequences for the existence of self-dual supercuspidal
representations of reductive $p$-adic groups.
