  We study Reeb dynamics on the three-sphere equipped with a tight contact form
and an anti-contact involution. We prove the existence of a symmetric periodic
orbit and provide necessary and sufficient conditions for it to bound an
invariant disk-like global surface of section. We also study the same questions
under the presence of additional symmetry and obtain similar results in this
case. The proofs make use of pseudoholomorphic curves in symplectizations. As
applications, we study Birkhoff's conjecture on disk-like global surfaces of
section in the planar circular restricted three-body problem and the existence
of symmetric closed Finsler geodesics on the two-sphere. We also present
applications to some classical Hamiltonian systems.
