We consider an inverse problem: Is there a backbone-like structure underlying the climate system? For this we propose a method to reconstruct and analyze a complex network from data generated by a spatio-temporal dynamical system. This technique is then applied to reanalysis and model surface air temperature data. Parameters of this network, as betweenness centrality, uncover relations to global circulation patterns in oceans and atmosphere. We especially study the role of hubs and of long range connections, called teleconnections, in the flows of energy and matter in the climate system. Next, we introduce a novel graph-theoretical framework for studying the interaction structure between sub-networks embedded within a complex network of networks. This allows us to quantify the structural role of single vertices or whole sub-networks with respect to the interaction of a pair of subnetworks on local, mesoscopic, and global topological scales. This concept is used to the climate system by introducing climate subnetworks representing different heights in the atmosphere. Parameters of this network of networks, as cross-betweenness, uncover relations to global circulation patterns in oceans and atmosphere. The global scale view on climate networks offers promising new perspectives for detecting dynamical structures based on nonlinear physical processes in the climate system.