  Fix a finite group $G$. We seek to classify varieties with $G$-action
equivariantly birational to a representation of $G$ on affine or projective
space. Our focus is odd-dimensional smooth complete intersections of two
quadrics, relating the equivariant rationality problem with analogous
Diophantine questions over nonclosed fields. We explore how invariants -- both
classical cohomological invariants and recent symbol constructions -- control
rationality in some cases.
