International audienceContext. Young giant planets and brown dwarf companions emit near-infrared radiation that can be linearly polarized up to several percent. This polarization can reveal the presence of an (unresolved) circumsubstellar accretion disk, rotation-induced oblateness of the atmosphere, or an inhomogeneous distribution of atmospheric dust clouds. Aims: We aim to measure the near-infrared linear polarization of 20 known directly imaged exoplanets and brown dwarf companions. Methods: We observed the companions with the high-contrast imaging polarimeter SPHERE-IRDIS at the Very Large Telescope. We reduced the data using the IRDAP pipeline to correct for the instrumental polarization and crosstalk of the optical system with an a...