  We present a scotogenic model, i.e. a one-loop neutrino mass model with dark
right-handed neutrino gauge singlets and one inert dark scalar gauge doublet
$\eta$, which has symmetries that lead to co-bimaximal mixing, i.e. to an
atmospheric mixing angle $\theta_{23} = 45^\circ$ and to a $CP$-violating phase
$\delta = \pm \pi/2$, while the mixing angle $\theta_{13}$ remains arbitrary.
The symmetries consist of softly broken lepton numbers $L_\alpha$ ($\alpha =
e,\mu,\tau$), a non-standard $CP$ symmetry, and three $Z_2$ symmetries. We
indicate two possibilities for extending the model to the quark sector. Since
the model has, besides $\eta$, three scalar gauge doublets, we perform a
thorough discussion of its scalar sector. We demonstrate that it can
accommodate a Standard Model-like scalar with mass $125\, \mathrm{GeV}$, with
all the other charged and neutral scalars having much higher masses.
