An unambiguous crystallographic structure solution for the observed phases II-VI of high pressure hydrogen does not exist due to the failure of standard structural probes at extreme pressure. In this work we propose that nuclear magnetic resonance spectroscopy provides a complementary structural probe for high pressure hydrogen. We show that the best structural models available for phases II, III, and IV of high pressure hydrogen exhibit markedly distinct nuclear magnetic resonance spectra which could therefore be used to discriminate amongst them. As an example, we demonstrate how nuclear magnetic resonance spectroscopy could be used to establish whether phase III exhibits polymorphism. Our calculations also reveal a strong renormalization of the nuclear magnetic resonance response in hydrogen arising from quantum fluctuations, as well as a strong isotope effect. As the experimental techniques develop, nuclear magnetic resonance spectroscopy can be expected to become a useful complementary structural probe in high pressure experiments.