The Hyuga embody the concepts of Taoism. The Rinnegan embodies the concepts of Budhism (and seems to have no clan - possibly being attainable by anyone who reaches an enlightened state, assuming Naruto keeps the Rinnegan in line with Budhism). The Uchiha embody the concepts of Shintoism (or, more so, its gods/goddesses).
Where the Senju and the Uzumaki fit in is a bit difficult to determine. Some of the concepts of the Senju seem mildly Korean while the Uzumaki somewhat Mongul - but that is shooting from the hip for me.
If one looks at the concepts within Taoism and how they relate to the way the Byakguan stacks up, already:
Figthing Styles:
Baguazhang, an internal martial art developed around the ideas of Taoism reflects the overall approach of the Gentle Fist and its varying techniques:
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The "Eight Trigrams" prefix to all of those techniques pretty much says it all.
But it goes deeper than that. T'ai chi ch'uan is another internal martial art designed around Taoism and seems to have been the influence for the chakra network and the Hyuga's ability to interupt the harmonious flow of energy in opponents or precisely manage it within themselves:
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And there's a third: Xingyiquan is yet another internal martial art developed around Taoism which seems to have developed from/around spear techniques; focusing on fast, powerful linear attacks inspired/named after animals:
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- this is exactly what we see within the very limited view we get of the Gentle Step: Twin Lion Fist (the only Hyuga art yet to mention an animal).
End Game Hyuga:
Uchiha gotta mind-****, Senju gotta heal, Uzumaki gotta chakra-'splode... what's a Hyuga to do? What is their 'niche' - their specialty and their ultimate state?
I think the answer to this lay in Taoism, as well.
The 'goal' of Taoism is to work within the Tao (nature) harmoniously ... to affect change without disrupting the natural order and/or expending great amounts of effort. The Chinese words for this were Wu and Wei. Wu is 'to be without' or 'to be empty' and Wei is 'to do' or as 'action/effort.' The end-goal of Taoism is "Wei Wu Wei" - or 'action without effort.'
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"It is in the state of Ming[citation needed] that the Taoist is in full harmony with Tao, and 'having arrived at this pointless point of non-action, there is nothing that is left undone.' It is upon achievement of this Chinese equivalent to 'enlightenment' that a sage begins to perform wei wu wei, or 'action without action.' Thus the sage will be able to work in harmony with Tao to accomplish what is needed, and, working in perfect harmony with the Tao, leave no trace of having done it."
The Over-Powered interpretation of that (also known as the Sharingan Viewpoint) means that a Hyuga at the peak of his/her ability makes Infinite Tsukuyomi look like a failed troll. Rather than creating the ultimate illusion - they change reality, itself, and do so without grand displays. They simply cause things to "work out" the way they need to.
The more reasonable interpretation of this (also known as the TenTen Viewpoint) is that the Hyuga at the peak of his/her abilities is able to directly effect nature chakra for spontaneous use. Rather than using the "gentle fist" on the "inner coils system" of people - they use it on the fabric of the universe, itself - using tiny pricks to cause other things to happen.
In short - the Hyuga were among the first sages (as we are familiar with them from the Toad Sage and other abilities).
I suspect it will be revealed that the Sage acquired the Rinnegan through a process of enlightenment (and that obtaining the Rinnegan is not genetics per se). But that the Hyuga and Uzumaki make up the paternal and maternal lineages of the sage, respectively.
It has been stated that the Rinnegan allows one to utilize any jutsu ... but that's not been seen in practice aside from claims that the Sage was responsible for starting the teaching of elemental ninjutsu to individuals.
That does, however, fit in with many of the concepts of the Byakugan. The Eight Trigrams are each the different 'states' of yin and yang within the universe (three 'slots' can be filled with either yin or yang chakra - which gives us 8 possible combinations). While Naruto seems to only deal with five of these elements (to be fair, Sky and Ocean seem to be a bit difficult to work in when Water, Wind, and Thunder/Lightning already exist...)... they all fall within the Hyuga's theme:
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So it's arguable that a Hyuga adept at manipulating natural chakra inherently gains the ability to mold nature chakra into an elemental form and utilize it (as well as detect the chakra affinities of others and instruct them on how to utilize these principles).
... Which, interestingly enough - the Uzumaki's Eight Trigrams seal falls within the same:
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It's quite possible that both the Hyuga and Uzumaki lineages predate the Sage - and that the Sage of Six Paths was 'merely' an Uzuamki endowed with a Byakugan who benefited from advanced Hyuga abilities (since lost or largely unutilized by the Hyuga) which enabled him to achieve enlightenment to unlock the Rinnegan.
His advanced ability to manipulate nature chakra into all elements led to the Rinnegan being incorrectly attributed with the ability to use all chakra natures. This also provides a branching of the Uzumaki/Senju at the sage (the Senju are the younger son while the Uzumaki are the maternal lineage of the Sage... although I would argue that, to fit more in line with known genetics - the 'younger son' would actually be the younger daughter) - with the Sage, himself, likely marrying to someone of the Uzumaki lineage (or the line that would become the Uzumaki). The strong interaction of the early Hyuga and Uzumaki created the advanced Uzumaki sealing system that would be passed down for generations to come.
The Hyuga persisted as normal, but the Sage's elder son fostered the Sharingan mutation.
Or something along those lines.
Though it's just my own thoughts on the matter.
