Context. Polarimetric imaging is one of the most effective techniques for high-contrast imaging and for the characterization of protoplanetary disks, and it has the potential of becoming instrumental in the characterization of exoplanets. The Spectro-Polarimetric High-contrast Exoplanet REsearch (SPHERE) instrument installed on the Very Large Telescope (VLT) contains the InfraRed Dual-band Imager and Spectrograph (IRDIS) with a dual-beam polarimetric imaging (DPI) mode, which offers the capability of obtaining linear polarization images at high contrast and resolution. Aims. We aim to provide an overview of the polarimetric imaging mode of VLT/SPHERE/IRDIS and study its optical design to improve observing strategies and data reduction. Methods. For H-band observations of TW Hydrae, we compared two data reduction methods that correct for instrumental polarization effects in different ways: a minimization of the "noise"image (Uφ), and a correction method based on a polarimetric model that we have developed, as presented in Paper II of this study. Results. We use observations of TW Hydrae to illustrate the data reduction. In the images of the protoplanetary disk around this star, we detect variability in the polarized intensity and angle of linear polarization that depend on the pointing-dependent instrument configuration. We explain these variations as instrumental polarization effects and correct for these effects using our model-based correction method. Conclusions. The polarimetric imaging mode of IRDIS has proven to be a very successful and productive high-contrast polarimetric imaging system. However, the instrument performance is strongly dependent on the specific instrument configuration. We suggest adjustments to future observing strategies to optimize polarimetric efficiency in field-tracking mode by avoiding unfavorable derotator angles. We recommend reducing on-sky data with the pipeline called IRDAP, which includes the model-based correction method (described in Paper II) to optimally account for the remaining telescope and instrumental polarization effects and to retrieve the true polarization state of the incident light.