Arena Fav Loops.
I'm only halfway to Duellist's Eye so i dunno about that one. I assume it loops too.
Arena Points loop. So do all your Fight Stats.
Blue Flower Hairpin loops. Pieces don't.
If you have the Looping Special Mission, it lists what you keep. Since the Mission is 0 stamina cost, check it out when it's available.
EDIT:
h43r:'d for detail.