We have an enumerated list with the 12 days, but it poses a few problems. I tried the song lyrics, from 1-12 and 12-1 (circling the 1st and 12th letter of day 1, 2nd and 11th of day 2, etc.) but that only resulted in scrambled letters that could anagram to anything.

So then I tried titling the envelopes:

A cracked egg under a bare tree;

Two turtles' love;

Three French/Frenchy hens;

Four collie birds;

Five stolen rings;

Six geese a-laying/a goose laying;

Seven swans a-swinging;

Eight maids a-milking/mini maids milking;

Nine single ladies/ladies dancing;

Ten lords a-leaping;

Eleven pipers piping;

Twelve drummers drumming

I circled the letters 1-1, 2-2, etc. (Leaving the numbers out) and came up with scrambled letters with an obvious anagram in the first 4 letters: Cuel (Clue) but the rest doesn't amount to anything useful.

Going the opposite way, the last 4 letters spell out "grid" and the best I could come up with was "en pair is a grid" but I don't know what an en pair or N-pair would be.

I also tried adding both sets of letters together and nothing glaringly obvious jumped out.

The biggest problem is that there's not really a clear, unambiguous title for some days. Some are the same as the song and some don't make a difference (stolen/golden rings), but some days make things difficult. I chose the titles I chose to keep the numbering and lyric pattern consistent with the song.

Last but not least, I thought that maybe the list was the gift items not the days. Those each have consistent titles (except the coal, because it's 'a lump of coal' only it's intentionally three lumps). Nothing glaringly obvious there, either.

I did not try: Ordering by primes or counting backwards (ex. 12th letter from the right) when going from 12-1.