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There are rules on how to pack dishes
in the square lacquer boxes.

(the first layer of lacquer ware)

Traditional dishes (including mitsuzakana) are packed in the ichinojyu. Sweet dishes and hors d'oeuvres are also packed in it.

(the second layer of lacquer ware)

Steamed dishes, boiled dishes and vinegared dishes are packed
in the ninojyu. The contents change for each home and generation.

(the third layer of lacquer ware)

Boiled dishes are packed in the sannojyu.

For an example of what might actually be packed. let's look at my mother's osechi-ryori.
Ichinojyu: mitsuzakana, tatakigobou, hinode-ebi, kamaboko , datemak‚‰ and kurikinton
Ninojyu: boiled eggs, grilled yellowtail, boiled lily root with eggs, namasu and baked chicken
Sannojyu: nimono and kobumaki.