  The correlations between different moments of two flow amplitudes, extracted
with the recently developed asymmetric cumulants, are measured in Pb$-$Pb
collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV recorded by the ALICE detector at
the CERN Large Hadron Collider. The magnitudes of the measured observables show
a dependence on the different moments as well as on the collision centrality,
indicating the presence of non-linear response in all even moments up to the
eighth. Furthermore, the higher-order asymmetric cumulants show different
signatures than the symmetric and lower-order asymmetric cumulants. Comparisons
with state-of-the-art event generators using two different parametrizations
obtained from Bayesian optimization show differences between data and
simulations in many of the studied observables, indicating a need for further
tuning of the models behind those event generators. These results provide new
and independent constraints on the initial conditions and transport properties
of the system created in heavy-ion collisions.
