  Determining the fundamental properties of the dark matter is one of the most
important open problems in particle physics today. If the dark matter particle
has spin zero, one of these properties is its CP nature. That is, whether it is
CP-even (scalar), CP-odd (pseudoscalar), or if its interactions violate CP. In
this paper, we show that the observation of $\gamma$-ray lines arising from the
decay of a spin-zero dark matter particle could be used to discriminate among
these possibilities. We consider a general setup where dark matter decay is
induced by effective operators and demonstrate that, due to gauge invariance,
there exists correlations among the branching ratios into gauge boson final
states ($\gamma\gamma$, $\gamma Z$, $W^+W^-$, $ZZ$) that depend on the mass and
the CP properties of the dark matter. Consequently, the future observation of
$\gamma$-ray lines may in principle be used to establish the CP nature of the
dark matter particle.
