  Given a quasi-split reductive group $G$ and a symmetric variety $X$, we
introduce a notion of endoscopic varieties for $(G,X)$, and establish the
foundational properties of these varieties such as matching of stable
semi-simple orbits. To do this, we introduce certain automorphism groups of
homogeneous spherical varieties, which encode the fine rational structure
needed to work over non-algebraically closed fields. In particular, we
establish the existence and uniqueness of the corresponding symmetric varieties
under a mild restriction of the characteristic of the field of definition. We
conjecture that this construction plays a role analogous to endoscopic groups
in the context of the relative trace formula. As evidence, we show how our
construction gives a pre-stabilization of regular elliptic terms of relative
trace formulae for many pairs $(G,X)$.
  When the cotangent bundle of the symmetric variety is hyperspherical, we
relate our theory to the Hamiltonian variety of the Langlands dual group
introduced by Ben-Zvi, Sakellaridis, and Venkatesh, proving some structural
conjectures for this variety in the symmetric setting.
