Turbulence and its organization, long conceptualized in terms of "coherent structures,"has resisted clear description. A significant limitation has been the lack of tools to identify instantaneous, spatially finite structures, while unraveling their superposition. We present a framework of generalized correlations, which can be used to readily define a variety of correlation measures, aimed at identifying field patterns. Coupled with Helmholtz-decomposition, this provides a paradigm to identify and disentangle structures. We demonstrate the correlations using vortex-based canonical flows and then apply them to incompressible, homogeneous, isotropic turbulence. We find that high turbulence kinetic energy (Ek) regions form compact velocity-jets that are spatially exclusive from high enstrophy (ω 2) regions that form vorticity-jets surrounded by swirling velocity. The correlation fields reveal that the energetic structures in turbulence, being invariably jets, are distinct from those in vortex-based canonical flows, where they can be jet-like as well as swirling. A full Biot-Savart decomposition of the velocity field shows that the velocity-jets are neither self-induced, nor induced by the interaction of swirling, strong vorticity regions, and are almost entirely induced, non-locally, by the permeating intermediate range (rms level) vorticity. Velocity-swirls, instead, are a superposition of self-induced and background-induced velocity. Interestingly, it is the mild intermediate vorticity that dominantly induces the velocity-field everywhere. This suggests that turbulence organization could result from non-local and non-linear field interactions, leading to an emergent description unlike the notion of a strict structural hierarchy. Our correlation-decomposition framework lends itself readily to the study of generic vector and scalar fields associated with diverse phenomena.@en