We report the oxygen K-edge spectra of ices Ill. VI, VII, and VIII measured with X-ray Raman Scattering. The pre-edge and main-edge contributions increase strongly with density, even though (lie hydrogen bond arrangements are very similar ill these phases. While the near-edge spectral features in water and ice have often been linked to hydrogen bonding, we show that the spectral changes in the phases studied here Call be quantitatively related to structural changes in the second coordination shell. Density-functional theory Calculations reproduce the experimental results and Support the conclusion. Our results Suggest that non-hydrogen-bonded neighbors can have a significant effect also in the liquid water spectrum. We discuss the implications of the results for the actively debated interpretation of the liquid water spectrum in terms of local structure.