  This paper studies the behavior of Jiu-Kang Yu's tame supercuspidal
representations relative to involutions of reductive p-adic groups. Symmetric
space methods are used to illuminate various aspects of Yu's construction.
Necessary conditions for a tame supercuspidal representation of G to be
distinguished by (the fixed points of) an involution of G are expressed in
terms of properties of the G-orbit of the associated G-datum. When these
conditions are satisfied, the question of whether a tame supercuspidal
representation is distinguished reduces to the question of whether certain
cuspidal representations of finite groups of Lie type are distinguished
relative to particular quadratic characters. As an application of the main
results, we obtain necessary and sufficient conditions for equivalence of two
of Yu's supercuspidal representations associated to distinct G-data.
