CP-odd invariants are useful for studying the CP properties of Lagrangians in
any basis. We explain how to build basis invariants for the scalar sector, and
how to distinguish CP-odd invariants from CP-even invariants. Up to a certain
order, we use these methods to systematically build all the CP-odd invariants.
The CP-odd invariants signal either explicit or spontaneous violation of CP.
Making use of the CP-odd invariants, we determine the CP properties of
potentials with 3 and with 6 Higgs fields arranged as triplets of specific
discrete symmetries in the $\Delta(3n^2)$ or $\Delta(6n^2)$ series (inlcuding
$A_4$, $S_4$, $\Delta(27)$ and $\Delta(54)$ as well as the cases for $n>3$).