We have performed the first broadband study of Mrk 421 from radio to TeV gamma rays with simultaneous measurements of the X-ray polarization from IXPE. The data were collected as part of an extensive multiwavelength campaign carried out between May and June 2022 using MAGIC, Fermi-LAT, NuSTAR, XMM-Newton, Swift, and several optical and radio telescopes to complement IXPE data. During the IXPE exposures, the measured 0.2-1TeV flux was close to the quiescent state and ranged from 25% to 50% of the Crab Nebula without intra-night variability. Throughout the campaign, the very high-energy (VHE) and X-ray emission are positively correlated at a 4{sigma} significance level. The IXPE measurements reveal an X-ray polarization degree that is a factor of 2-5 higher than in the optical/radio bands; that implies an energy-stratified jet in which the VHE photons are emitted co-spatially with the X-rays, in the vicinity of a shock front. The June 2022 observations exhibit a rotation of the X-ray polarization angle. Despite no simultaneous VHE coverage being available during a large fraction of the swing, the Swift-XRT monitoring reveals an X-ray flux increase with a clear spectral hardening. This suggests that flares in high synchrotron peaked blazars can be accompanied by a polarization angle rotation, as observed in some flat spectrum radio quasars. Finally, during the polarization angle rotation, NuSTAR data reveal two contiguous spectral hysteresis loops in opposite directions (clockwise and counterclockwise), implying important changes in the particle acceleration efficiency on approximately hour timescales.