  In multi-component scalar dark matter scenarios, a single $Z_N$ ($N\geq 4$)
symmetry may account for the stability of different dark matter particles. Here
we study the case where $N$ is even ($N=2n$) and two species, a complex scalar
and a real scalar, contribute to the observed dark matter density. We perform a
phenomenological analysis of three scenarios based on the $Z_4$ and $Z_6$
symmetries, characterizing their viable parameter spaces and analyzing their
detection prospects. Our results show that, thanks to the new interactions
allowed by the $Z_{2n}$ symmetry, current experimental constraints can be
satisfied over a wider range of dark matter masses, and that these scenarios
may lead to observable signals in direct detection experiments. Finally, we
argue that these three scenarios serve as prototypes for other two-component
$Z_{2n}$ models with one complex and one real dark matter particle.
