If I'm identifying all those icons correctly, your helmets are sorted by descending ID #:

Snowman Head 6968
Aetherial Mythril Celata 2813
Cavalry Mask 2772
Aetherial Steel Celata 2756
Torturer's Monocle 2748

Aetherial Silver Circlet (Heliodor) 2742 (The gem may be different; there's a set of sequential numbers here.)
Velveteen Bandana 2727
Aetherial Reinforced Iron Sallet 2726
Aetherial Iron Elmo 2724
Doctore's Chain Coif 2721

Gridanian Soldier's Cap 2715
--- I stopped going from the top down here and did a few from the bottom up ---
Dream Hat 2642
Helm of Light 2634
Moogle Cap 2633

Most likely, "category" is worthless for the armory chest, because it's already divided by category (helms, gloves, etc.), and when it doesn't find anything else to sort by, it goes by ID #.