  The most general Higgs potential of the two-Higgs-doublet model (2HDM)
contains three squared-mass parameters and seven quartic self-coupling
parameters. Among these, one squared-mass parameter and three quartic coupling
parameters are potentially complex. The Higgs potential explicitly violates CP
symmetry if and only if no choice of basis exists in the two-dimensional Higgs
``flavor'' space in which all the Higgs potential parameters are real. We
exhibit four independent potentially complex invariant (basis-independent)
combinations of mass and coupling parameters and show that the reality of all
four invariants provides the necessary and sufficient conditions for an
explicitly CP-conserving 2HDM scalar potential. Additional potentially complex
invariants can be constructed that depend on the Higgs field vacuum expectation
values (vevs). We demonstrate how these can be used together with the
vev-independent invariants to distinguish between explicit and spontaneous
CP-violation in the Higgs sector.
