  The change in the configuration of valence protons between the initial and
final states in the neutrinoless double-$\beta$ decay of $^{130}$Te
$\rightarrow$ $^{130}$Xe and of $^{136}$Xe $\rightarrow$ $^{136}$Ba has been
determined by measuring the cross sections of the ($d$,$^3$He) reaction with
101-MeV deuterons. Together with our recent determination of the relevant
neutron configurations involved in the process, a quantitative comparison with
the latest shell-model and interacting-boson-model calculations reveals
significant discrepancies. These are the same calculations used to determine
the nuclear matrix elements governing the rate of neutrinoless double-$\beta$
decay in these systems.
