  A long standing conjecture in Hamiltonian Dynamics states that every contact
form on the standard contact sphere $S^{2n+1}$ has at least $n+1$ simple
periodic Reeb orbits. In this work, we consider a refinement of this problem
when the contact form has a suitable symmetry and we ask if there are at least
$n+1$ simple symmetric periodic orbits. We show that there is at least one
symmetric periodic orbit for any contact form and at least two symmetric closed
orbits whenever the contact form is dynamically convex.
