Flavor Anisotropy in the High-Energy Astrophysical Neutrino Sky

High-energy astrophysical neutrinos, with TeV--PeV energies, offer unique insight into astrophysics and particle physics. Their incoming directions and flavor composition -- i.e., the proportion of electron, muon, and tau neutrinos in their flux -- are, individually, rewarding observables. Combined, they offer new opportunities, hitherto unexplored, that we expose for the first time. Anisotropy in the arrival directions of electron, muon, and tau neutrinos may reveal multiple populations of neutrino sources, differently distributed in the sky, and test whether neutrinos of different flavor propagate preferentially along certain directions, such as expected from breaking Lorentz invariance. Using 7.5 years of public IceCube High-Energy Starting Events, we make the first measurement of the directional flavor composition of high-energy astrophysical neutrinos, constrain the presence of flavor dipoles and quadrupoles, and improve constraints on "compass asymmetries" introduced by Lorentz invariance violation. In the near future, upcoming neutrino telescopes will improve these measurements across the board.