Context. The diverse morphology among protoplanetary disks may result from planet-disk interactions, suggesting the presence of planets undergoing formation. The characterization of disks can provide information on the formation environments of planets. To date, most imaging campaigns have probed the polarized light from disks, which is only a fraction of the total scattered light and not very sensitive to planetary emission. Aims. We aim to observe and characterize protoplanetary disk systems in the near-infrared in both polarized and total intensity light to carry out an unprecedented study of the dust scattering properties of disks, as well as of any possible planetary companions. Methods. Using the star-hopping mode of the SPHERE instrument at the Very Large Telescope, we observed 29 young stars hosting protoplanetary disks and their reference stars in the K_s-band polarized light. We extracted disk signals in total intensity by removing stellar light using the corresponding reference star observations, by adopting the data imputation concept with sequential non-negative matrix factorization (DI-sNMF). For well-recovered disks in both polarized and total intensity light, we parameterized the polarization fraction phase functions using a scaled beta distribution. We investigated the empirical DI-sNMF detectability of disks using logistic regression. For systems with SPHERE data in the Y, J, and H bands, we have summarized their polarized color at an approximately 90° scattering angle. Results. We obtained high-quality disk images in total intensity for 15 systems and in polarized light for 23 systems. The total intensity detectability of disks primarily depends on the host star brightness, which determines adaptive-optics control ring imagery and thus stellar signals capture using DI-sNMF. The peak of polarization fraction tentatively correlates with the peak scattering angle, which could be reproduced using certain composition for compact dust, yet more detailed modeling studies are needed. Most of the disks are blue in polarized J – K_s color and the fact that they are relatively redder as stellar luminosity increases indicates larger scatterers. Conclusions. High-quality disk imagery in both total intensity and polarized light allows for disk characterizations in the polarization fraction. Combining these techniques reduces the confusion between the disk and planetary signals.