  We further develop and simplify the general theory of distinguished tame
supercuspidal representations of reductive $p$-adic groups due to Hakim and
Murnaghan, as well as the analogous theory for finite reductive groups due to
Lusztig. We apply our results to study the representations of ${\rm GL}_n(F)$,
with $n$ odd and $F$ a nonarchimedean local field, that are distinguished with
respect to an orthogonal group in $n$ variables. In particular, we determine
precisely when a supercuspidal representation is distinguished with respect to
an orthogonal group and, if so, that the space of distinguishing linear forms
has dimension one.
