We further develop and simplify the general theory of distinguished tame supercuspidal representations of reductive
p
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
G
L
n
(
F
)
\mathrm {GL}_n(F)
, with
n
n
odd and
F
F
a nonarchimedean local field, that are distinguished with respect to an orthogonal group in
n
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.