  The XY pyrochlore antiferromagnet Er$_2$Ti$_2$O$_7$ exhibits a rare case of
$Z_6$ discrete symmetry breaking in its $\psi_2$ magnetic ground state. Despite
being well-studied theoretically, systems with high discrete symmetry breakings
are uncommon in nature and, thus, Er$_2$Ti$_2$O$_7$ provides an experimental
playground for the study of broken $Z_n$ symmetry, for $n>2$. A recent
theoretical work examined the effect of a magnetic field on a pyrochlore
lattice with broken $Z_6$ symmetry and applied it to Er$_2$Ti$_2$O$_7$. This
study predicted multiple domain transitions depending on the crystallographic
orientation of the magnetic field, inducing rich and controllable
magnetothermodynamic behavior. In this work, we present neutron scattering
measurements on Er$_2$Ti$_2$O$_7$ with a magnetic field applied along the [001]
and [111] directions, and provide the first experimental observation of these
exotic domain transitions. In a [001] field, we observe a $\psi_2$ to $\psi_3$
transition at a critical field of 0.18$\pm$0.05T. We are thus able to extend
the concept of the spin-flop transition, which has long been observed in Ising
systems, to higher discrete $Z_n$ symmetries. In a [111] field, we observe a
series of domain-based phase transitions for fields of 0.15$\pm$0.03T and
0.40$\pm$0.03T. We show that these field-induced transitions are consistent
with the emergence of two-fold, three-fold and possibly six-fold Zeeman terms.
Considering all the possible $\psi_2$ and $\psi_3$ domains, these Zeeman terms
can be mapped onto an analog clock - exemplifying a literal clock anisotropy.
Lastly, our quantitative analysis of the [001] domain transition in
Er$_2$Ti$_2$O$_7$ is consistent with order-by-disorder as the dominant ground
state selection mechanism.
