Considering the two days as a single holiday, or as two separate sanctities: Eruvin 39a, 39b-40a; Beitzah 3a-b, 4b-5b
Treating the two days as a single holiday even after the enactment of Rabbi Yochanan ben Zakkai, lest people err when the rule returns to accept testimony only until midday: Beitzah 5b
Separating a tithe and saying that it will take effect on whichever of the two days is not actually the holiday: Eruvin 39a, 39b; Beitzah 3a-b
Prayers
Mentioning in Rosh HaShanah prayers, "If this is the right day, answer me today. Otherwise, answer me on the other day.": Eruvin 39a, 40a