  We present a precision analysis of the $^{136}$Xe two-neutrino $\beta\beta$
electron spectrum above 0.8 MeV, based on high-statistics data obtained with
the KamLAND-Zen experiment. An improved formalism for the two-neutrino
$\beta\beta$ rate allows us to measure the ratio of the leading and subleading
$2\nu\beta\beta$ nuclear matrix elements (NMEs), $\xi^{2\nu}_{31} =
-0.26^{+0.31}_{-0.25}$. Theoretical predictions from the nuclear shell model
and the majority of the quasiparticle random-phase approximation (QRPA)
calculations are consistent with the experimental limit. However, part of the
$\xi^{2\nu}_{31}$ range allowed by the QRPA is excluded by the present
measurement at the 90% confidence level. Our analysis reveals that predicted
$\xi^{2\nu}_{31}$ values are sensitive to the quenching of NMEs and the
competing contributions from low- and high-energy states in the intermediate
nucleus. Since these aspects are also at play in neutrinoless $\beta\beta$
decay, $\xi^{2\nu}_{31}$ provides new insights towards reliable neutrinoless
$\beta\beta$ NMEs.
