Madara is Natsume's grandfatherGoing off on Natsume's mother being half-youkai, and Reiko not having any humans to have a child with...maybe this is the reason Madara always has Reiko mentioned to him, and why he continues to protect Natsume. He wants to look after his grandson.
Reiko's mother was a kitsune.As you all know, Kitsune are powerful female (Li'l Kitsune is Special) fox spirits who will sometimes mate with humans when in human form. They are also known to be tricksters, as most foxes are, but are known to keep promises. Now, let's look at Reiko for a moment. Reiko is a young girl who has great spiritual power, and often challenges spirits to duels. She oftentimes uses trickery in order to win them, and has a mischievous personality overall. Those who lose her duels are forced to become her underlings, but she never tries to force the victor to do the same, just as she says. Add this to the fact that she has gold, slit-pupiled eyes, and it's easy to see how this could be possible.
Madara is Reiko's father.Other youkai often mention Reiko to him and imply that he is only interested in Natsume because of Reiko.
Natsume's father was half youkai.Seriously, how Reiko came to have a child is never explained, people didn't like her, and she was always alone. How could she have possibly have had a human partner?
Natsume is somehow related to GacktGackt once claimed that he could see spirits, and that his family didn't believe him and began to suspect that he was mentally unstable (Granted, Gackt says a lot of ridiculous things, as the 496-year-old vampire claim demonstrated, but still). An bishonen that saw dead people/youkai and was considered disturbed because of it? Gee, now why does that sound familiar...?
Natsumes parents were killed by youkaiI'm probably overthinking this but, Matoba said former exorcist clans have trouble with youkai even long after the last person who could see was born into them. Reiko was somebody, who would probably draw the groudge of a lot of youkai to her. Natsume himself is an orphan. It is possible.
Matoba's right eye would have been targeted after he graduated from high schoolIn fact, there's a picture of him in high school and his right eye is not covered. While this is just bonus material, his right eye also seems to be different (although it could simply be inconsistent art), meaning that he became the clan head after his graduation.
Natsume Reiko lost the ability to see youkai soon after high school.Whenever we see her in a youkai's flashbacks, she is always wearing her school uniform. It was even stated at some point that many people lose the ability to see youkai when they grow up. Fridge Horror, since if it's true, the same may happen to Natsume Takashi.
Natori is Natsume's amazingly-young-looking grandpa, on his mother's side.Because not only can he see youkai, but he looks just. Like him. And he can be very sadistic especially about youkai, just like Reiko. Seriously, he and Reiko would make the perfect match. Whereas Reiko hated most humans, she would probably have gotten along with a powerful (and fabulous) exorcist. He may even know about it, since he seems to have paternal instincts towards Takashi and even offered to adopt him. A rival exorcist even suggested that Natori is taking jobs away from other exorcists in the town where "Natsume Reiko's grandson" lives suggesting that he certainly knows who Reiko is and probably met her while she was alive.
Natsume Yuujinchou is an AU of Gunnerkrigg Court
Takashi is Reiko's reincarnation....As well as her grandson. Thus why he is always mistaken for her by youkai, and why he has been having flashbacks/dreams about her in recent chapters. It may be that he has her soul, not just a physical resemblance to her. VERY strongly hinted at when, at the end of his dream, a small youkai grabs his wrist and insists "You ARE Reiko."
Madara knew/was close with Natsume's grandfatherMadara said earlier in the series that the human forms he can shift into are Reiko, Takashi (though this achieved later on in the series) and an old man. This troper thinks that old man is possibly Natsume's grandfather. Madara had to know the person well enough in order to shape-shift into them so they had to be close somehow.
Nishimura has some spiritual powerThough probably to the same extent as Taki's. There are several instances in the manga which Nishimura hears Nyanko-sensei (though Natsume usually brushes it off as something else). Notably, in the special chapters that featured a bit on his past, Nishimura was able to hear Nyanko-sensei when he was knocked out (though he thought it was a dream). Nyanko-sensei said that for someone to hear him needs to have some spiritual powers though it doesn't not have to be as much as Natsume's (like Tanuma or Taki). This might lead to in future chapters that when Natsume's friends do find out about Natsume's ability, Nishimura might point that out.
Despite Natori's orders, Natori's shiki won't tell him about the Book of FriendsNyanko-sensei states in one chapter that ayakashi have a unspoken code to not speak about certain things they don't want to be known like the Book of Friends. Even if they are Natori's shiki and Urihime and Sasago kind of don't like Natsume, they won't directly tell Natori what Natsume holds. Of course, that doesn't mean Natori won't somehow find out elsewhere (since Natori keeps hearing about the book's existence through the slip of the tongue of other ayakashi).
Natori's gecko tattoo is trying to sell insuranceAnd he won't leave until you spend 15 minutes to save 15% on your car insurance.
Madara used to be a common dog and was sealed as a punishmentAccording to Japanese folklore (aka Wikipedia), an inugami is created by burying a dog, separating its head from its body, and then taking it to a shrine. Whoever evoked Madara wanted him to commit crimes but some exorcist probably sealed him before he could.