  The dissipation of energy flux in blazar jets plays a key role in the
acceleration of relativistic particles. Two possibilities are commonly
considered for the dissipation processes, magnetic reconnection -- possibly
triggered by instabilities in magnetically-dominated jets -- , or shocks -- for
weakly magnetized flows. We consider the polarimetric features expected for the
two scenarios analyzing the results of state-of-the-art simulations. For the
magnetic reconnection scenario we conclude, using results from global
relativistic MHD simulations, that the emission likely occurs in turbulent
regions with unstructured magnetic fields, although the simulations do not
allow us to draw firm conclusions. On the other hand, with local
particle-in-cell simulations we show that, for shocks with a magnetic field
geometry suitable for particle acceleration, the self-generated magnetic field
at the shock front is predominantly orthogonal to the shock normal and becomes
quasi-parallel downstream. Based on this result we develop a simplified model
to calculate the frequency-dependent degree of polarization, assuming that
high-energy particles are injected at the shock and cool downstream. We apply
our results to HBLs, blazars with the maximum of their synchrotron output at
UV-soft X-ray energies. While in the optical band the predicted degree of
polarization is low, in the X-ray emission it can ideally reach 50\%,
especially during active/flaring states. The comparison between measurements in
the optical and in the X-ray band made during active states (feasible with the
planned {\it IXPE} satellite) are expected to provide valuable constraints on
the dissipation and acceleration processes.
