We investigate local high chirality inside a microcavity near exceptional points (EPs) achieved via asymmetric backscattering by two internal weak scatterers. At EPs, coalescent eigenmodes exhibit position-dependent and symmetric high chirality characteristics for a large azimuthal angle between the two scatterers. However, asymmetric mode field features appear near EPs. Two azimuthal regions in the microcavity classified by the scatterers exhibit different wave types and chirality. Such local mode field features are attributed to the symmetries of backscattering in direction and spatial distribution. The connections between the wave types, the symmetry of mode field distribution and different symmetries of backscattering near EPs are also analyzed and discussed. Benefiting from the small size of weak scatterers, such microcavities with a high Q/V near EPs can be used to achieve circularly polarized quantum light sources and explore EP modified quantum optical effects in cavity quantum electrodynamics systems.