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.

30 pages, 12 figures. Major revision. This version was accepted for publication in Physical Review C