The extraordinary DIBs observed toward Herschel 36 (Dahlstrom et al. 2013) have been analyzed (Oka et al. 2013). The analysis led us to a new way to classify the carriers of DIBs depending on whether the molecules are polar or non-polar. The pronounced Extended Tails toward Red (ETR) observed for DIBs λ5780.5, λ5797.1, and λ6613.6 are explained as due to radiative excitation of high rotational levels of polar carrier molecules in an environment with high radiative temperature ~90 K. Other DIBs (e.g., λ5849.8, λ6196.0, and λ6379.3) which do not show ETR are likely due to non-polar molecules. Model calculations taking into account the interplay of radiative and collisional effects reproduce the observed ETR using realistic molecular parameters if the radiative temperature is sufficiently high (~90 K). The calculation suggests that the carriers of DIBs with ETR are likely medium size molecules with 3 - 6 heavy atoms unless the radiative temperature is much higher.