  Let $X$ be a nonempty real variety that is invariant under the action of a
reflection group $G$. We conjecture that if $X$ is defined in terms of the
first $k$ basic invariants of $G$ (ordered by degree), then $X$ meets a
$k$-dimensional flat of the associated reflection arrangement. We prove this
conjecture for the infinite types, reflection groups of rank at most $3$, and
$F_4$ and we give computational evidence for $H_4$. This is a generalization of
Timofte's degree principle to reflection groups. For general reflection groups,
we compute nontrivial upper bounds on the minimal dimension of flats of the
reflection arrangement meeting $X$ from the combinatorics of parabolic
subgroups. We also give generalizations to real varieties invariant under Lie
groups.
