I extrapolated a good deal, based on canon, and thought it would be well to post an overview of the result.

Canon

Canonically, we know that there are multiple branches of the Fuuchouin clan and style. There is the main house, the Fuuchouin, also referred to as the "omote", the front or face. There are the Kokuchouin, also referred to as the "ura", the back or hidden. There are the Western and Eastern branches. We know that the family name associated with the Eastern branch is Toufuuin. The family names are written as follows, translating 風 (fuu) as "grace" for ease of reading.

風鳥院 (Fuuchouin): House of the Bird of Paradise, or, more literally, of the Graceful Bird.

黒鳥院 (Kokuchouin): House of the Black Swan, or, more literally, of the Black Bird.

東風院 (Toufuuin): Eastern House of Grace.

We also know that, during Kazuki and Saizou’s fights, both as children and as adults, they use techniques named the Dance of the Red Bird and the Dance of the Green Dragon, respectively.

We know that the Houses have hallmarks to their style, that the main house focuses on beauty and grace, that the Western style focuses on strength, and that the Eastern style focuses on speed. (see Voodoo Child 14)

We know that the Phoenix is a secret technique, handed down among the women who marry into the Fuuchouin main house.

Extrapolation

Names

Based on the pattern of names above, I have assumed that the family name associated with the Western Fuuchouin will be 西風院 (Seifuuin), the Western House of Grace.

Four Beasts, Four Houses

The Green Dragon and the Red Bird are two of the four cardinal guardian beasts found in Chinese astrology and myth and imported into Japanese literature. The Green Dragon is the beast of the East and the Red Bird the beast of the South. Based on this, and on the characters who use those two techniques, I have made two assumptions. One is that there are two other Dances within the Fuuchouin clan repertoire: the Dance of the White Tiger (the western beast) and the Dance of the Black Tortoise (the northern beast). The other is that the four Houses of the clan are symbolically and perhaps philosophically aligned with a beast and direction to each. The Eastern Fuuchouin are seen to use the Green Dragon. This suggests that the main house, represented by Kazuki using the Red Bird, is aligned with the South.

If this is so, it gives the Western Fuuchouin the White Tiger, and places the Kokuchouin in the North with the Black Tortoise.

Ura and the North

Based on the assumption above, and the hallmarks of the three styles that are noted in canon, I also extrapolated that the hidden techniques of the black strings were not the original hallmark of the Kokuchouin. The hallmarks of the other three have some relation to their symbolic beasts. The Red Bird of the South is a creature of elegance and grace, for example. The astrological associations of the tiger and dragon (metal and water) have led to them becoming a common metaphor in the martial arts for straightforward attack versus the indirect, which lines up reasonably well with strength versus speed.

Thus, the mark of the Kokuchouin style should be something having to do with the associations of the Black Tortoise. These are longevity, stability and wisdom; winter; and water. Based on this, I made the supposition that the hallmark of the northern style is endurance. Following this logic, the Kokuchouin must, at one point, have been an integral part of the Fuuchouin circle of Houses and had their own regular scrolls and techniques. The techniques of the black string, then, must have been, at first, not associated with any one House, and only placed in the keeping of the Kokuchouin later, thus turning them into the ura, or hidden House and apparently displacing their traditional style.

Four Principles

This is total extrapolation, based on the foregoing. I needed to figure out what the four Dances might actually do, so I decided that, logically, they should express the spirit of their quarter and particular style. It’s hard to imagine an attack that expresses grace or endurance, though, so I reached a little further. The techniques we see in canon seemed to me to divide into four basic functions: bindings that capture, strings that cut like Rain Shower, strings that strike like Comet, and deflections like Jade Shield. I aligned these with the four hallmarks of the Houses, resulting in: East/speed/cut, South/grace/deflect, West/strength/strike, North/endurance/bind. Thus I assume that the Dance of the Green Dragon is a powerful cutting technique while the Red Bird is an equally powerful deflection, which explains why they cancelled each other out both times.

Dragon and Phoenix

Now the fun part. First of all, the Red Bird of the South is not the Phoenix. Those are two different mythological entities. The Phoenix is a more inclusive creature, showing all five of the cardinal colors, and has been used to represent the Empress during imperial periods.

About those five colors. There are actually five guardian beasts that can show up, to go with the five elements of the Taoist cosmology. The fifth is earth, in the center. This is associated with the Yellow Dragon, who is used to represent the Emperor.

Thus, for a truly united clan, there should be a fifth Dance, and the nature of that technique should match the association of the element of earth: unity. The principle I chose to go with this Dance was, in a way, the first one Kazuki mentions; when he fights Akame, the thing that allows him to win is his understanding that it’s the vibration imparted to the strings that gives them strength. Thus, the principle that runs under and unites the other four is vibration or resonance. In accordance with my assumption that the Fuuchouin lost their balance, their unity and harmony as four Houses, when they assigned the Kokuchouin to the hidden techniques, I also posited that the Kachoufuugetsu, the ultimate technique that expresses the Fuuchouin’s valuation of beauty above all, was something that replaced this earlier Dance. Originally, my theory goes, what the heir to the clan had to demonstrate was his understanding of unity and balance, not simply of beauty. The conflict was probably embedded in the clan’s origin, of course, given that the main house, the source of the entire clan, is named for the Bird of the South, the graceful and elegant bird. The primacy of the southern aspects simply swung too far and unbalanced the whole.

When the center has returned and the Dragon and the Phoenix are united, in the persons of the couple who lead the entire clan, they should produce perfect harmony.

Invented Techniques

There are only a few string techniques named in canon, really, so I had to invent some more.

Butterfly: Similar to Flower Dance though not as powerful, this technique sends strings slipping on sidelong vectors, like the flight of a butterfly, to get through a straightforward defense or attack.

Mountain Lake: An advanced defensive technique that reflects, rather than simply blocking, the opponent’s attack, as perfectly as the water of a still lake reflects an image.

Sunrise: In the best tradition of Getbackers Physics, this technique sets strings to reflect the available light at the opponent, blinding them to one’s movements. Sunrise in Spring is the particularly Toufuuin variation, which does so very quickly, creating a flash more than a glare.

Flowing River: Deflects an attack upward, just as a river swirls at the bends, and strikes through the space thus created like the fast current in the river’s center. (This is essentially Yang style’s Fair Lady, only with strings.)

Winter Gale: This is a two-pronged attack; even if it does not bind, or freeze, the opponent it is meant to blind them with a wall of strings, as with a heavy snowfall, so they can’t attack accurately.

Blossoming Plum: This is an offensive technique meant to be used while under attack, just as the plum blossoms while the winter frosts are still hanging on; it takes momentum from the opponent’s attack to strike more strongly.

Night Forest Web: This is a defensive technique designed to entangle the opponent’s strings, as one is entangled in the spider webs spun at night while walking through a forest.