Stokes profiles emerging from the magnetized solar photosphere and observed by SOT/SP aboard the HINODE satellite exhibit a variety of complex shapes. These are indicative of unresolved magnetic structures that have been overlooked in the inversion analyses performed so far. Here we present the first interpretation of the Stokes profile asymmetries measured in the ion{Fe}{i} 630 nm lines by SOT/SP, in both quiet Sun internetwork (IN) and network regions. The inversion is carried out based on the hypothesis of MIcro-Structured Magnetized Atmosphere (MISMA), where the unresolved structure is assumed to be optically thin. We analyze a 29.52 arcsec × 31.70 arcsec subfield carefully selected to be representative of the properties of a 302 arcsec × 162 arcsec quiet Sun field-of-view (FOV) at the disk center. The inversion code is able to reproduce the observed asymmetries in a very satisfactory way, including 35% of the inverted profiles with large asymmetries. The inversion code interprets 25% of inverted profiles as emerging from pixels in which both positive and negative polarities coexist. These pixels are located in either frontiers between opposite polarity patches or very quiet regions. The kG field strengths are found at the base of the photosphere in both network and IN regions; in the case of the latter, both kG fields and hG fields are admixed. When considering the magnetic properties of the mid photosphere, most kG fields do not exist, and the statistics is dominated by hG fields. According to the magnetic filling factors derived from the inversion, we constrain the magnetic field of only 4.5% of the analyzed photosphere (and this percentage reduces to 1.3% when considering all pixels, including those with low polarization that have not been analyzed). The properties of the rest of the plasma imply that weak fields do not contribute to the detected polarization signals. The average flux densities derived in the full subfield and IN regions are higher than those derived from the same dataset by Milne-Eddington (ME) inversion. We detect large asymmetries in the HINODE SOT/SP polarization profiles. These are not negligible in quiet Sun data. The MISMA inversion code reproduces them in a satisfactory way, and provides a statistical description of the magnetized IN and network which partly differs and complements the results obtained so far. The importance of having a complete interpretation of the line profile shapes is therefore clearly evident.