  The detection of the high-energy ($\sim290$ TeV) neutrino coincident with the
flaring blazar TXS 0506+056, the first and only $3\sigma$ neutrino-source
association to date, provides new, multimessenger tests of the weak equivalence
principle (WEP) and Lorentz invariance. Assuming that the flight time
difference between the TeV neutrino and gamma-ray photons from the blazar flare
is mainly caused by the gravitational potential of the Laniakea supercluster of
galaxies, we show that the deviation from the WEP for neutrinos and photons is
conservatively constrained to have an accuracy of $10^{-6}-10^{-7}$, which is
3--4 orders of magnitude better than previous results placed by MeV neutrinos
from supernova 1987A. In addition, we demonstrate that the association of the
TeV neutrino with the blazar flare sets limits on the energy scales of quantum
gravity for both linear and quadratic violations of Lorentz invariance (LIV) to
$E_{\rm QG, 1}>3.2\times10^{15}-3.7\times10^{16}$ GeV and $E_{\rm QG,
2}>4.0\times10^{10}-1.4\times10^{11}$ GeV. These improve previous limits on
both linear and quadratic LIV energy scales in neutrino propagation by 5--7
orders of magnitude.
