Debris disks give us the unique opportunity to probe the properties of small mu m-sized particles, allowing us to peer into the constituents of their parent bodies, young analogs of comets and asteroids of our solar system. In the past, studies of the total intensity phase function, the brightness of the disk as a function of the scattering angle, have proven powerful to constrain the main characteristics of the dust particles in debris disks. Nonetheless, there can remain some degeneracies in the modeling that can be alleviated when considering polarized intensity observations. We obtained new near-infrared scattered light observations of four young debris disks, and used state-of-the-art algorithms to recover the total intensity and linear polarimetric images of the disks. These images allow us to constrain the degree of linear polarization as a function of the scattering angle. All four debris disks are detected in polarized intensity, and three are also recovered in total intensity. We measured a peak degree of polarization of <~40% for all three disks. For the disk around HD 129590, we are also able to determine the degree of polarization in the radiation-pressure-driven halo. To reproduce the observed polarization fractions, we find that the particles must consist of highly refractive and absorbing material. For HD 129590, by measuring the polarization fraction beyond the birth ring, we constrain the width of the size distribution to be increasingly small toward greater radii, which is compatible with the effect of radiation pressure. We put these findings to the test and present a self-consistent approach to producing synthetic images, assuming different profiles for the radiation pressure strength, and accounting for the presence of unbound grains. We find the contribution of these grains to be especially critical in order to reproduce the increasing degree of polarization with stellocentric distance. Some of our results - namely a very small blow-out size and very large (n, k) values for the optical constants, which are required to reproduce the observed degree of polarization - might seem difficult to reconcile with our understanding of cosmic dust. Similar results have been obtained for other disks and we discuss the current limitation of available light-scattering models as well as possible avenues to alleviate these limitations.