  The main theme of this paper is the dynamics of Reeb flows with symmetries on
the standard contact sphere. We introduce the notion of strong dynamical
convexity for contact forms invariant under a group action, supporting the
standard contact structure, and prove that in dimension $2n+1$ any such contact
form satisfying a condition slightly weaker than strong dynamical convexity has
at least $n+1$ simple closed Reeb orbits. For contact forms with antipodal
symmetry, we prove that strong dynamical convexity is a consequence of ordinary
convexity. In dimension five or greater, we construct examples of antipodally
symmetric dynamically convex contact forms which are not strongly dynamically
convex, and thus not contactomorphic to convex ones via a contactomorphism
commuting with the antipodal map. Finally, we relax this condition on the
contactomorphism furnishing a condition that has non-empty $C^1$-interior.
