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CP-odd invariants provide a basis independent way of studying the CPproperties of Lagrangians. We propose powerful methods for constructing basisinvariants and determining whether they are CP-odd or CP-even, thensystematically construct all of the simplest CP-odd invariants up to a givenorder, finding many new ones. The CP-odd invariants are valid for generalpotentials when expressed in a standard form. We then apply our results toscalar potentials involving three (or six) Higgs fields which form irreducibletriplets under a discrete symmetry, including invariants for both explicit aswell as spontaneous CP violation. The considered cases include one triplet ofStandard Model (SM) gauge singlet scalars, one triplet of SM Higgs doublets,two triplets of SM singlets, and two triplets of SM Higgs doublets. For eachcase we study the potential symmetric under one of the simplest discretesymmetries with irreducible triplet representations, namely $A_4$, $S_4$,$\Delta(27)$ or $\Delta(54)$, as well as the infinite classes of discretesymmetries $\Delta(3n^2)$ or $\Delta(6n^2)$.

CP-odd invariants for multi-Higgs models: applications with discrete symmetry