Context. Polarimetric imaging is one of the most effective techniques for high-contrast imaging and characterization of protoplanetary disks, and has the potential to be instrumental in characterizing exoplanets. VLT/SPHERE contains the InfraRed Dual-band Imager and Spectrograph (IRDIS) with a dual-beam polarimetric imaging (DPI) mode, which offers the capability to obtain linear polarization images at high contrast and resolution. Aims. We aim to provide an overview of IRDIS/DPI and study its optical design to improve observing strategies and data reduction. Methods. For H-band observations of TW Hya, we compare two data reduction methods that correct for instrumental polarization effects in different ways: a minimization of the noise image, and a polarimetric-model-based correction method that we present in Paper II of this study. Results. We use observations of TW Hya 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 with pointing-dependent instrument configuration. We explain these variations as instrumental polarization effects and correct for these effects using our model-based correction method. Conclusions. IRDIS/DPI has proven to be a very successful and productive high-contrast polarimetric imaging system. However, the instrument performance depends on the specific instrument configuration. We suggest adjustments to future observing strategies to optimize polarimetric efficiency in field tracking mode by avoiding unfavourable derotator angles. We recommend reducing on-sky data with the pipeline called IRDAP that 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.
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