The Eight Days of Christmas/New Years (No, not THOSE 8 days)

       Language is odd sometimes, but then it's also interesting.

       I thought that recently when I realized "The Eight Days of Christmas/New Years":

       There's Christmas Eve and Christmas Day; and there's New Year's Eve and New Year's Day.  But none of these is either Christmas Eve day, or New Year's Eve day; nor, for that matter, Christmas Day eve or New Years Day eve!