  We present the generalization of recently introduced observables for the
studies of correlated fluctuations of different anisotropic flow amplitudes,
dubbed Symmetric Cumulants. We introduce a new set of higher order observables
and outline a unique way how the genuine multi-harmonic correlations can be
extracted from multi-particle azimuthal correlators. We argue that correlations
among flow amplitudes can be studied reliably with the general mathematical
formalism of cumulants only if that formalism is applied directly on the flow
amplitudes. We have tested all the desired properties of new observables with
the carefully designed Toy Monte Carlo studies. By using the realistic
iEBE-VISHNU model, we have demonstrated that their measurements are feasible
and we have provided the predictions for the centrality dependence in Pb--Pb
collisions at LHC energies. A separate study was performed for their values in
the coordinate space. The new observables contain information which is
inaccessible to individual flow amplitudes and correlated fluctuations of only
two flow amplitudes, and therefore they provide further and independent
constraints for the initial conditions and the properties of Quark-Gluon Plasma
in high-energy nuclear collisions.

Nuclear Experiment;High Energy Physics - Experiment;Nuclear Theory