Context. The combination of high-contrast imaging with spectroscopy and polarimetry offers a pathway to studying the grain distribution and properties of debris disks in exquisite detail. Here, we focus on the case of a gas-rich debris disk around HD 141569A, which features a multiple-ring morphology first identified with SPHERE in the near-infrared. Aims. We obtained polarimetric differential imaging with SPHERE in the H-band to compare the scattering properties of the innermost ring at 44 au with former observations in total intensity with the same instrument. In polarimetric imaging, we observed that the intensity of the ring peaks in the south-east, mostly in the forward direction, whereas in total intensity imaging, the ring is dete...