Authors:Asensio-Torres et alAbstract:We present high-contrast angular differential imaging (ADI) observations of the debris disk around HD 32297 in H-band, as well as the first polarimetric images for this system in polarized differential imaging (PDI) mode with Subaru/HICIAO. In ADI, we detect the nearly edge-on disk at >5sigma levels from ~0.45 arcsec to ~1.7 arcsec (50-192 AU) from the star and recover the spine deviation from the midplane already found in previous works. We also find for the first time imaging and surface brightness (SB) indications for the presence of a gapped structure on both sides of the disk at distances of ~0.75 arcsec (NE side) and ~0.65 arcsec (SW side). Global forward-modelling work delivers a best-fit model disk and well-fitting parameter intervals that essentially match previous results, with high-forward scattering grains and a ring located at 110 AU. However, this single ring model cannot account for the gapped structure seen in our SB profiles. We create simple double ring models and achieve a satisfactory fit with two rings located at 60 and 95 AU, respectively, low-forward scattering grains and very sharp inner slopes. In polarized light we retrieve the disk extending from ~0.25-1.6 arcsec, although the central region is quite noisy and high S/N are only found in the range ~0.75-1.2 arcsec. The disk is polarized in the azimuthal direction, as expected, and the departure from the midplane is also clearly observed. Evidence for a gapped scenario is not found in the PDI data. We obtain a linear polarization degree of the grains that increases from ~10% at 0.55 arcsec to ~25% at 1.6 arcsec. The maximum is found at scattering angles of ~90degrees, either from the main components of the disk or from dust grains blown out to larger radii.