A large number of water models exist for molecular simulations. They differ in the ability to reproduce specific features of real water instead of others, like the correct temperature for the density maximum or the diffusion coefficient. Past analysis mostly concentrated on ensemble quantities, while few data were reported on the different microscopic behavior. Here, we compare seven widely used classical water models (SPC, SPC/E, TIP3P, TIP4P, TIP4P-Ew, TIP4P/2005, and TIPP) in terms of their local structure-forming capabilities through hydrogen bonds for temperatures ranging from 210 to 350 K by the introduction of a set of order parameters taking into account the configuration of up to the second solvation shell. We found that all models share the same structural pattern up to a temperature shift. When this shift is applied, all models overlap onto a master curve. Interestingly, increased stabilization of fully coordinated structures extending to at least two solvation shells is found for models that are able to reproduce the correct position of the density maximum. Our results provide a self-consistent atomic-level structural comparison protocol, which can be of help in elucidating the influence of different water models on protein structure and dynamics.