The astrophysical neutrinos recently discovered by the IceCube neutrino telescope have the highest detected neutrino energies -from TeV to PeV -and travel the longest distances -up to a few Gpc, the size of the observable Universe. These features make them naturally attractive probes of fundamental particle-physics properties, possibly tiny in size, at energy scales unreachable by any other means. The decades before the IceCube discovery saw many proposals of particle-physics studies in this direction. Today, those proposals have become a reality, in spite of prevalent astrophysical unknowns. We showcase examples of studying fundamental neutrino physics at these scales, including some of the most stringent tests of physics beyond the Standard Model.1 Based on an invited talk given at the 6th Symposium on Prospects in the Physics of Discrete Symmetries