  We analyze CP symmetry in symplectic modular-invariant supersymmetric
theories. We show that for genus $g\ge 3$ the definition of CP is unique, while
two independent possibilities are allowed when $g\le 2$. We discuss the
transformation properties of moduli, matter multiplets and modular forms in the
Siegel upper half plane, as well as in invariant subspaces. We identify
CP-conserving surfaces in the fundamental domain of moduli space. We make use
of all these elements to build a CP and symplectic invariant model of lepton
masses and mixing angles, where known data are well reproduced and observable
phases are predicted in terms of a minimum number of parameters.
