  The existence and spatio-temporal patterns of $2\pi$-periodic solutions to
second order reversible equivariant autonomous systems with commensurate delays
are studied using the Brouwer $O(2) \times \Gamma \times \mathbb
Z_2$-equivariant degree theory. The solutions are supposed to take their values
in a prescribed symmetric domain $D$, while $O(2)$ is related to the reversal
symmetry combined with the autonomous form of the system. The group $\Gamma$
reflects symmetries of $D$ and/or possible coupling in the corresponding
network of identical oscillators, and $\mathbb Z_2$ is related to the oddness
of the right-hand side. Abstract results, based on the use of Gauss curvature
of $\partial D$, Hartman-Nagumo type {\it a priori bounds} and Brouwer
equivariant degree techniques, are supported by a concrete example with $\Gamma
= D_8$ -- the dihedral group of order $16$.
