Weakly magnetized neutron stars in X-ray binaries show a complex phenomenology with several spectral components that can be associated with the accretion disk, the boundary, and/or a spreading layer, a corona, and a wind. Spectroscopic information alone, however, is not enough to distinguish these components. The analysis of the timing data revealed that most of the variability, and in particular, kilohertz quasi-period oscillations, are associated with the high-energy component that corresponds to the boundary and/or spreading layer. Additional information about the nature of the spectral components, and in particular, about the geometry of the emission region, can be provided by X-ray polarimetry. One of the objects of the class, a bright, persistent, and rather peculiar galactic Type I X-ray burster GX 13+1, was observed with the Imaging X-ray Polarimetry Explorer (IXPE) and the XMM–Newton. Using the XMM–Newton data, we obtained the best-fit values for the continuum spectral parameters and detected strong absorption lines associated with the accretion disk wind. IXPE data showed the source to be significantly polarized in the 2–8 keV energy band, with an overall polarization degree (PD) of 1.4%±0.3% at a polarization angle (PA) of −2° ±6° (errors at the 68% confidence level). During the two-day long observation, we detected rotation of the PA by about 70° with the corresponding changes in the PD from 2% to nondetectable and then up to 5%. These variations in polarization properties are not accompanied by visible spectral state changes of the source. The energy-resolved polarimetric analysis showed a significant change in polarization, from being strongly dependent on energy at the beginning of the observation to being almost constant with energy in the later parts of the observation. As a possible interpretation, we suggest a constant polarization component, strong wind scattering, or a different polarization of the two main spectral components with an individually peculiar behavior. The rotation of the PA suggests a misalignment of the neutron star spin from the orbital axis.