Long way back in history
There existed a 4th major clan, Kuzunoha (or Abe) clan.
Abe no Seimei established this clan, in the Izumi Province of Shinoda forest, years later (when Tsuchimikado Family was already a famous and powerful branch of the Abe clan), to show gratitude to his late mother— Lady Kuzunoha. Even though she left when he was around 5, but the stories by his dad stayed with him later in life. Part guilt because he was, in a way, the reason his mother had to leave.
They had isolated themselves for centuries after the Heian era because of the heating political environment and to conserve their energy after warding off the Vengeful Spirit, Sugawara no Michizane.
The head of the clan is currently a woman named Kaguya Kuzunoha, who manifested the clan's inherited technique named Illusion.
Now, the relation of the Kuzunoha clan ancestors/ and present with others:-
Gojo clan (branch of Sugawara clan) :-
In the Heian hierarchy, both clans were considered "technical" officials rather than pure high-nobles like the Fujiwara.
The Sugawara were the masters of Literature and History. They ran the university system.
The Abe were the masters of Onmyodo (Divination, Astronomy, and Magic).
They worked side-by-side in the Imperial Palace. If the Emperor needed a poem written, he called a Sugawara; if he needed a curse lifted or a calendar made, he called an Abe.
They were in the same system, but in different departments.
Later in history, both clans eventually married into the Fujiwara clan differently, in different eras. Because the Fujiwara acted as a massive "genetic hub," the blood of the Sugawara and the blood of the Abe eventually met in the middle.
By the late Heian period, many high-ranking nobles could claim Sugawara no Michizane as an ancestor on one side and Abe no Seimei (or other prominent Abe) on the other.
Hence, the distant distant cousin bond.
Hence, Gojo Clan and Kuzunoha Clan serves a distant cousin relation. Over time, the Gojo clan grew closer to the Kuzunoha branch than to the Kamo or Zenin.
And eventually, they were referred as "Sibling clans" or Siblings in Bond. Gojo clan provided them protection from politics, Kuzunoha clan provided them seals for specific purposes.
Kamo clan:-
Historically, During Heian Era(the golden age of sorcery),the Kamo family - specifically Kamo no Tadayuki and his son Kamo no Yasunori were the official teachers of Abe no Seimei. However, Seimei (son of Abe no Yasuna and the fox-spirit Kuzunoha) became so talented that he eventually surpassed his teachers.
Abe descendants still looked at Kamos as their teachers, respects and gratitude. And Kamo looked at Abe like their masterpiece.
But, when Tsuchimikado Family (branch of Abe clan founded by Abe no Seimei) got divided into another small branch named Kuzunoha Clan. Kamo (after death of kamo no yasunori) felt a deep sense of irritation. The mastery of the responsibilities (and cursed energy and techniques in JJK terms) given by Kamo, the natural genius, all went to a forest, far from politics, it felt like an insult to the Great Kamo.
The remaining Tsuchimikado family was now combining non sorcerers and sorcerers who don't want to continue the sorcery.
This deeply offended Kamo, their Star student— The genius Seimei descendants—were either not continuing, or inside a deep forest.
Though Kuzunoha and Abe clans see Kamo clan as their teacher, with respect and gratitude. But Kamo sees the Kuzunoha clan as a waste of the potential, soon later, they became envious and irritated (the later generation), because Kuzunoha still had impact even though they disconnected from the political world and leaved behind most of the Orthodox traditions.
Zenin Clan:-
In real Japanese history, the warrior class (which the Zenin represent) and the Onmyoji (spiritual advisors like the Abe/Kuzunoha line) often clashed for the Emperor's favor. Zenins represent the rise of the Samurai dealing with strength, weapons, and seizing power by force. Meanwhile kuzunoha represents the Heian-era aristocracy including rituals, poetry, and influencing the world through unseen spirits. They were/are Political Enemies. The Zenin try to suppress the Kuzunoha's influence in the modern era, every time, calling their methods "outdated superstitions," while the Kuzunoha would look down on the Zenin as "thugs with Cursed Energy."
The Zenin value Results and Might. Their techniques (like the Ten Shadows or Projection Sorcery) are often about physical dominance and tactical superiority. To them, a true sorcerer is a warrior.
The Abe/Kuzunoha value Lineage and Ritual. Their power comes from ancient contracts, Shikigami like the Twelve Heavenly Generals, and complex Onmyōdō.
The Zenin clan mirrors the Taira or Minamoto clans (the warriors), while The Kuzunoha clan mirrors the Fujiwara or Abe (the court nobles).