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 degrees +/- 6 degrees (errors at the 68% confidence level). During the two-day long observation, we detected rotation of the PA by about 70 degrees 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.