As there are two Kamidogu Clans, they both possess different philosophies with only one core value being the same between them.
The Kamidogu of the Moon's philosophy is that the clan is the primary family and loyalty to the clan is paramount above all things. They are closed off in regards to outsiders interfering in clan issues or politics, and teach this mentality to all members from birth. This way of life often leads to the clan being alienated by the village, which does cause some friction in regards to village politics.
The Kamidogu of the Sun's philosophy is almost the polar opposite in regards to how integrated the clan is with the Sun village. They believe that the clan is simply one piece of the puzzle that makes the village what it is. While they do take pride in their clan, they take just as much if not more in their village. They are essentially an open book and more than willing to share their culture and ideals with outsiders.
It's these differences that caused the clan to split into two, with hatred or disdain carrying on with both sides for years. The only value that the two actually agree on is that all members respect and value the traditions of the clan. This is why both clans still follow the old ways in regards to dreading the hair and valuing the philosophies of the clan; despite there being two.
Location: Sun and Moon (Two Warring Sects)
The Kamidogu clan had once been a single powerful clan, dating as far back as when Higoru walked the lands. At this time they were beholden to no village and lived secretly in the the mountainous regions; only coming to villages for trade. After the great battle with the Faceless One, the Kamidogu returned home after allying themselves with the Sun in order to assist in said battle. It was then that the issues within the clan began. Having fought alongside so many other people and clans, one side wanted to become more open to the world. The other side wanted to continue to live in secret and stick to the more traditional values.
The issue became over a desire to progress alongside the rest of the world versus a desire to maintain what the clan has already established. What began as conversation quickly progressed to violence and resulted in a war between the two sides. This infighting resulted in the clan splitting with the traditionalist moving toward the territory of the Moon. Communication was severed and afterwards the two sides shared only the use of smoke jutsu, with any encounter between the sides usually resulting in fighting.
Over time the two Sects became somewhat foreign to each other in regards to what would go on within the clan. Each instead focusing more on their respective Sects, as both would eventually be integrated into the world. Even the Moon Kamidogu sect chose this by means of survival, though still maintained an air of secrecy that often gave them an air of mystery in the village. Something that is maintained to the current date. The Kamidogu of the Sun are much more integrated into society and are much more welcomed among the village.
To this day both Sects of the clan still very much hate each other, with some even speculating that the hatred stems from something more than a simple disagreement on traditions. Regardless, much of the more in depth details of the clans history is unknown as both sects only carry a pieces of the truth. And the bad blood, it is unlikely this truth will ever be known.