  Let $F/F_{\mathsf{o}}$ be a quadratic extension of non-archimedean locally
compact fields of odd residual characteristic and $\sigma$ be its non-trivial
automorphism. We show that any $\sigma$-self-dual cuspidal representation of
${\rm GL}_n(F)$ contains a $\sigma$-self-dual Bushnell--Kutzko type. Using such
a type, we construct an explicit test vector for Flicker's local Asai
$L$-function of a ${\rm GL}_n(F_{\mathsf{o}})$-distinguished cuspidal
representation and compute the associated Asai root number. Finally, by using
global methods, we compare this root number to Langlands--Shahidi's local Asai
root number, and more generally we compare the corresponding epsilon factors
for any cuspidal representation.
