LkCa 15 hosts a pre-transitional disk as well as at least one accreting protoplanet orbiting in its gap. Previous disk observations have focused mainly on the outer disk, which is cleared inward of similar to 50 au. The planet candidates, on the other hand, reside at orbital radii around 15 au, where disk observations have been unreliable until recently. Here, we present new J-band imaging polarimetry of LkCa 15 with SPHERE IRDIS, yielding the most accurate and detailed scattered-light images of the disk to date down to the planet-hosting inner regions. We find what appear to be persistent asymmetric structures in the scattering material at the location of the planet candidates, which could be responsible at least for parts of the signals measured with sparse-aperture masking. These images further allow us to trace the gap edge in scattered light at all position angles and search the inner and outer disks for morphological substructure. The outer disk appears smooth with slight azimuthal variations in polarized surface brightness, which may be due to shadowing from the inner disk or a two-peaked polarized phase function. We find that the near-side gap edge revealed by polarimetry matches the sharp crescent seen in previous ADI imaging very well. Finally, the ratio of polarized disk to stellar flux is more than six times larger in the J-band than in the RI bands.