While the mechanisms driving flares in blazar jets remain poorly understood, the associated spectral variations imply that particle (re)acceleration must play a central role. For HSPs, the advent of sensitive X-ray polarimetry allows a direct probe of the magnetic field structure impacting the most energetic particles in the jet, leading to unprecedented insights into the most extreme particle acceleration mechanisms. As leptonic scenarios predict that X-rays are emitted by particles that also produce TeV photons, the combination of very high-energy (VHE; $E>0.1$ TeV) and X-ray polarimetry data is of prime importance to further constrain the physical origin of flares.

In December 2023 we observed an outburst of the archetypal TeV HSP Mrk 421 from radio to VHE gamma rays with MAGIC, Fermi-LAT, Swift, XMM-Newton, and several optical and radio telescopes. Over a two-week period, a simultaneous characterization of the X-ray polarization was obtained thanks to the Imaging X-ray Polarimetry Explorer (IXPE), in addition to optical and radio polarimetry data. We find substantial variability in both X-rays and VHE gamma rays, with the highest VHE flux occurring during the IXPE observing window, and exceeding twice the flux of the Crab Nebula. The average X-ray polarization degree is significantly higher than that at radio and optical frequencies, suggesting an energy stratification of the emitting region. Notably, the X-ray/VHE flux changes are accompanied by strong variability of the X-ray polarization angle and degree on timescales of days. This behavior points toward a significant impact of plasma turbulence on the origin of X-ray/VHE flux variability. The highest X-ray polarization degree reaches 26%, around which an X-ray counter-clockwise hysteresis loop is measured, implying that the ~keV emission originates from the high-energy cutoff of the particle population. We model the broadband emission with a simplified stratified jet model throughout the flare using daily binned SEDs. Our analysis favors a scenario in which the flare is produced by a turbulent plasma crossing a shock front at which particles get accelerated.