Purim and Yom Kippur

Rabbi Hutner (Pachad Yitzchak Purim 22) quotes from Vilna Gaon that Purim and Yom Kippur are not just linked, but are two halves of the same day. We paskin that Yom Tov is to be Chatzi Lashem and Chatzi Lachem, half for Hashem and half for you. Vilna Gaon suggests that Yom Kippur is the half of the Yom Tov which is Lashem while Purim is the half of the Yom Tov which is Lachem.
Rabbi Hutner explains that Yom Kippur is the day we were saved from Hashem’s decree to destroy the Jews after the sin of the Golden Calf (Devarim 9:14), while Purim is the day we were saved from Haman’s decree to destroy the Jews (Esther 3:13). Both decrees are described with the Hebrew root SHeMaD, and thus they are linked. The difference is that the decree we were saved from on Yom Kippur was heavenly while the decree we were saved from on Purim was earthly. Therefore the expression of Kedusha on those days represents the respective decrees from which we were saved on them, with Yom Kippur being a day of Kedusha which is Lashem (heavenly) and Purim being a day of Kedusha which is Lachem (earthly).
Rabbi Hutner explains that this manifests itself in the interpersonal aspects of the Holidays as well. On both Yom Kippur and Purim we have obligations to other Jews. On Yom Kippur the interpersonal aspects are spiritual, we have to appease others for wrongs we have done them. It is a connection of the spirit because the day is Lashem. On Purim we send others manos, food. It is a gift of the physical, because the day is Lachem.
Purim and Yom Kippur, two halves of the same Holiday.