  The MeV spectral peak of gamma-ray bursts (GRBs) is best explained as
photospheric emission from a dissipative relativistic jet. The observed
non-blackbody spectrum shows that sub-photospheric dissipation involves both
thermal plasma heating and injection of nonthermal particles, which quickly
cool through inverse Compton scattering and emission of synchrotron radiation.
Synchrotron photons emitted around and above the photosphere are predicted to
dominate the low-energy part of the GRB spectrum, starting from roughly a
decade in energy below the MeV peak. We show that this leads to a unique
polarization signature: a rise in GRB polarization toward lower energies. We
compute the polarization degree of GRB radiation as a function of photon energy
for a generic jet model, and show the predictions for GRBs 990123, 090902B and
110721A. The expected polarization is significant in the X-ray band, in
particular for bursts similar to GRB 090902B. Radiation in the MeV peak (and at
higher energies) is unpolarized as long as the jet is approximately uniform on
angular scales $\delta\theta \gtrsim \Gamma^{-1}$ where $\Gamma$ is the bulk
Lorentz factor of the jet.
