Ok, I will try to explain the deal behind Kyo and Iori rivalry. It is NOT easy.
Please bear with my terrible English and with the many typos as well. I have no time at all to proofread the whole thing. Sorry.
To understand KOF plot, you have first to know a famous Japanese myth, part of the Kojiki traditional mythology book.
In the Kojiki, we have (among others) two great deities. Amaterasu Omikami, goddess on Sun and creation, and his brother Susano-oh, god of chaos and destruction.
At a certain point of the story, they had a fight and a bad quarrel.
Ok, now there was a huge beast, an eight-headed serpent/dragon called Yamata no Orochi. This beast requested a virgin maiden to be sacrificed to him each year.
One year, the selected maiden was princess Kushinada. Actually, she was Amaterasu's beloved one vestal too. Thus, Susano-oh though to go and kill Orochi, save the princess and thus re-join with his offended sister by saving her beloved one.
Therefore, Susano-oh did. He went to Orochi dressed up as a kind of wanderer merchant, and got the beast drunk using eight jar s of sake (each one for one of the Orochi heads). Then, he cut all of Orochi's head one by one and killed the beast. On the very Orochi tail, Susano-oh finds a sword, which was named Kusanagi [literally 'grass slayer', as dragons were believed to live beneath grass]. The chaos god kept the Kusanagi sword and presented it to the Kushinada family, along with a mystic mirror, made from one of the Orochi blood tear, and a mystic bead, know as 'Magatama', created with one of the Orochi tooth.
The three Sacred Regalia were kept by the Kushinada family, and became the three sacred objects of Japanese tradition. You can find them in a Tokyo temple, and references to this famous myth are sprouted all around in Japan culture (anime included, just think of Sailor Moon S).
Now, keep in mind this myth is believed to happen before the beginning of time, about 1800 years ago.
King of Fighters storyline move by a reinterpretation of this myth.
In the SNK version of the myth, Orochi was not really a serpent, but a clan of fighters whose blood was marked by the 'Orochi Power'. They were eight warriors, as the Eight Heads of the allegorical serpent. They were the Hakkeshu [literally 'the eight mighty'], and they also asked eight human (virgins) to be sacrificed in order to resurrect the Orochi itself, the virgins were called 'the Kushinada'.
About 1800 years ago, three clans dared to fight the Orochi. They were the Kusanagi, the Yasakani and the Yata family. Both Kusanagi and Yasakani possessed the amazing fighting ability to generate and thrust holy flames. The Yata family was a Shinto priest family. Each family represents one of the three mystic regalia. Yasakani, 'the Magatama bead' froze the serpent, Kusanagi 'the sword' defeated it and Yata 'the mirror' sealed off the Orochi. Since then, the Yata family always had a lineage of twin sisters in which blood bonds consisted the seal of the Orochi.
Century passed. The evil Orochi was defeated, but its clan survived. After about 1180 years, the Yasakani clan became blinded by greed and ambition, and started an internal fight with the Kusanagi clan. The Yasanaki ended up being seduced by the Orochi clan and power.
The Yasakani mixed up its blood and lineage with the Orochi clan, gaining great power. Since then, the Yasakani flames turned from crimson to purple, and all the Yasakani got shortened lives since the inhuman power flowing in their veins was consuming them. At that time, the Yasakani changed their name in YAGAMI.
Yes, this was the very beginning of the so-called '660 years war' between the Kusanagi and Yagami clans, later families.
Kyo and Iori are obviously the descendants of these families.
The Yata family, not to be involved in the fight and in order to prevent the Orochi seal to be removed, changed its name as well to keep secret. They became the KAGURA family. Yes, Chizuru is the descendant of the Yata/Kagura clan and family.
The story arrives this way to our day.
Iori is an 'Orochi bastard', he thrust purple flame and get pain by that.
In modern times, the modern Hakkeshu are about to gather.
The stronger of them, Goenitz, tries three missions. First, he goes to the only traitor Hakkeshu, called Guidel. This Hakkeshu was shouting away his Orochi blood living a normal life with a normal family. When Guidel refuses to rejoin with Goenitz, Goenitz awake the half-Orochi blood of the little Guidel daughters. That little girl was LEONA. She go berserk by 'Riot of the Blood', which affect the Orochi bastard when they half-Orochi blood awakes. Leona kill their parents and fall in catatonic state, being later saved by HEIDERN, but remembering nothing of the experience.
Later, Goenitz manage to kill Chizuru Kagura twin sister, thus breaking the Orochi seal.
A man then tried to siphon some of the Orochi power: he was RUGAL. Goenitz was curios to see how the Orochi power would affect a 'normal' human, so let Rugal live, only taking his right eye as a blood mark. Goenitz also send his servants, two Hakkeshu as well, to spy Rugal. They were Mature and Vice, which played the role of Rugal secretaries in order to survey him.
In the end, Rugal was consumed by the Orochi power and died (KOF95).
In 96, Goenitz revealed himself, just to be defeated by the joined forces of the three sacred family Kusanagi (Kyo), Yasakani/Yagami (Iori) and Yata/Kagura (Chizuru). In this occasion, Leona re-gained part of her memories, and also Iori Yagami suffered his first Riot of the Blood, which lead him killing his two teammates Vice and Mature, which were also sent by Goenitz to spy Yagami and later become traitor of their master.
Goenitz, the Hakkeshu of the Wind, died.
Mature and Vice died as well.
Guidel was dead since years.
In 1997, three of the four remaining Hakkeshu awakened and tried to resurrect the Orochi, The were the New Faces Team, Chris of the Flames, Shermi of the Thunder and Yashiro of the Earth.
The eighth Hakkeshu was Yamazaki Ryuji.
The NFT tried to seized the last Kushinada remaining, to kill her and resurrect the Orochi. She actually was YUKI, Kyo's girlfriend.
In the end, after both Leona and Iori suffered Riot of the Blood by the upcoming release of the Orochi, the NFT sacrificed themselves and the Orochi was reborn in Chris body, but again defeated by the three family.
Now I guess you can understand way in 97/98 Iori as a 'freezing' super, Chizuru has a 'sealing' super, and Kyo has a smashing super.
You will also know why Chizuru turns into a Shinto priest when facing the Hakkeshu, or why she draw a mirror in the air while facing Yagami or Kusanagi.
You would also understand why we have 'berserker' version of Leona and Iori in 97, or why Yamazaki react to Chris presence.
There are many, many more details, but I guess I have already written too much.
I hope someone enjoyed all this.
