  The neutron-rich $^{213}$Pb isotope was produced in the fragmentation of a
primary 1 GeV $A$ $^{238}$U beam, separated in FRS in mass and atomic number,
and then implanted for isomer decay $\gamma$-ray spectroscopy with the RISING
setup at GSI. A newly observed isomer and its measured decay properties
indicate that states in $^{213}$Pb are characterized by the seniority quantum
number that counts the nucleons not in pairs coupled to angular momentum $J=0$.
The conservation of seniority is a consequence of the Berry phase associated
with particle-hole conjugation, which becomes gauge invariant and therefore
observable in semi-magic nuclei where nucleons half-fill the valence shell. The
$\gamma$-ray spectroscopic observables in $^{213}$Pb are thus found to be
driven by two mechanisms, particle-hole conjugation and seniority conservation,
which are intertwined through the Berry phase.
