  We consider the impact of imposing generalized CP symmetries on the Higgs
sector of the two-Higgs-doublet model, and identify three classes of
symmetries. Two of these classes constrain the scalar potential parameters to
an exceptional region of parameter space which respects either a Z_2 discrete
flavor symmetry or a U(1) symmetry. We exhibit a basis-invariant quantity that
distinguishes between these two possible symmetries. We also show that the
consequences of imposing these two classes of CP symmetry can be achieved by
combining Higgs Family symmetries, and that this is not possible for the usual
CP symmetry. We comment on the vacuum structure and on renormalization in the
presence of these symmetries. Finally, we demonstrate that the standard CP
symmetry can be used to build all the models we identify, including those based
on Higgs Family symmetries.
