We discuss various results pertaining to the possible (minimal) target dimensions of rational sphere maps given various symmetries. We analyze completely what happens when the Hermitian equivariant group (introduced by D'Angelo-Xiao) is the Unitary group, we discuss the gaps for the possible invariant groups (necessarily cyclic), and we discuss an optimization problem whose solution in dimensions 3 or more leads to a symmetrized version of the Whitney map. Precise formulas for the maps in the two-dimensional version of this problem seem intractable, but we provide considerable information about them.