  A mechanism to construct asymptotically flat, isolated, stationary black hole
(BH) spacetimes with no $\mathbb{Z}_2$ (No$\mathbb{Z}$) isometry is described.
In particular, the horizon geometry of such No$\mathbb{Z}$ BHs does not have
the usual north-south (reflection) symmetry. We discuss two explicit families
of models wherein No$\mathbb{Z}$ BHs arise. In one of these families, we
exhibit the intrinsic horizon geometry of an illustrative example by
isometrically embedding it in Euclidean 3-space, resulting in an "egg-like"
shaped horizon. This asymmetry leaves an imprint in the No$\mathbb{Z}$ BH
phenomenology, for instance in its lensing of light; but it needs not be
manifest in the BH shadow, which in some cases can be analytically shown to
retain a $\mathbb{Z}_2$ symmetry. Light absorption and scattering due to an
isotropic source surrounding a No$\mathbb{Z}$ BH endows it with a non-zero
momentum, producing an asymmetry triggered BH rocket effect.
