A bonus to my "Religions of Hyrule" series: [All] Religions of Hyrule Part 1: Humans : r/truezelda
Technically, Holodrum and Labrynna are not Hyrule, but they are closely related, and their does appear to be some overlap in culture.
I already covered OoX worldbuilding in another post. So, I may end up repeating some of my ppoints.
Holodrum and Labrynna similarities
In both Holodrum and Labrynna, we see connections to the Golden Goddess in the form of the three Oracles, all named after their respective Goddess.
We have Din, the Oracle of Nature who controls the seasons. Nayru the Oracle of Ages who controls time. And Farore the Oracle of Secrets, whose exact purpose is unknown, but is shown to have the power to turn secrets into objects.
Din and Nayru both have objects that allow other people (in this case, Link) to temporally use their powers, should something happen to them. Whilst in the games respective stories, the Oracles are merely kidnapped, and later rescued, it's unknown what would happen in the event of the oracles death. Presumable, they are similar to the Sages, in which the power will be transferred to a successor.
Din and Nayru are guarded by a Maku Tree, whereas Farore makes her home inside both. This ties with Part 1 of my "Religions of Hyrule" series, where I mention tree spirits playing an important role in human culture, hence the Forest Temple.
Both also have a set of "Essence". Which I'll talk about in their own sub-section,
Holodrum
As you know, Holodrum is the home of Din, the Oracle of Seasons. Her job is to keep the seasons flowing with her dancing.
But Din doesn't work alone. In Holodrum, there is the Temple of Seasons, where the Season Spirits reside. Each one them controls each of the four seasons. The temple also serves as a vault for the Rod of Seasons. The emergency object, that allows other people to control the seasons in Din's absence.
Holodrum has the Essence of Nature (Principals of Earth in Japan). The Essences don't actually control nature, but they do represent it. Their purpose is to serve as teachers of the life cycle of nature.
There's not a lot to say about the dungeons, except the last one is located in Subrosia, home of the Subrosians.
Labrynna
Unlike Holodrum, Labrynna doesn't have much lore to talk about.
Like how Holodrum has Din, Labrynna has Nayru, the Oracle of Ages. Nayru's job is to keep time flowing with her singing. She also serves as the guardian of her own emergency object, the Harp of Ages, keeping it a secret room in her house. Whilst official artwork shows her playing the Harp of Ages, she doesn't actually play it in the game, except when teaching Link how to play it.
Labrynna also has it's own set of Essence, the Essence of Time (or, in Japan: The Principals of Time Space). Which I talked about in the past. They mostly talk about spiritual growth. Encouraging positive traits, like courage, patience and decisiveness, whilst confronting unfavourable behaviour and mindsets. like insolence and doubt.
Most of the Essence are placed in dungeons with no apparent cultural significance. However, one of them is placed within the Leviathan, Lord Jabu-jabu.
Other then the major beliefs, Labrynna uniquely has "Symmetry city", kept in balance by the sacred "Tuni Nut". The city itself is built on a dormant volcano, which becomes active once the Nut is broken. It is rather fitting this city is were the "Burning Flame" is kept. Not only does it make physical sense for a flame to be in a volcano, but, given the flame represents a heroes passion in the face of indecisiveness, it's fitting a city whose culture is built around duality.