Magatsu-Izanagi (禍津伊邪那岐) means "filth-tarnished/Calamity Haven Izanagi". This refers to Izanagi's state before he washed away the filth from Yomotsu Hirasaka in a failed mission to retrieve his late consort, Izanami.
Magatsu-Izanagi is the Persona of Tohru Adachi, the one responsible for the murders in Inaba. It follows the form of Izanagi, as it is actually bestowed upon Adachi without Adachi's knowledge by Izanami when he first arrived in the rural town. Magatsu-Izanagi takes the form of a bloody Izanagi, and its stats and skills form Adachi's battle stats.
Although Adachi doesn't seem to represent any Arcana in the original release of Persona 4, he actually encompasses the reversed Fool Arcana: impulsiveness, poor judgement, obsession and frivolity. Adachi acted on impulse when he murdered Mayumi Yamano and Saki Konishi because they both weren't interested in him. He shows poor judgement when he underestimated the protagonist and his group during their confrontation. Additionally, he was obsessed with keeping his "cat-and-mouse" game alive, doing whatever it takes to ensure the murders continue, such as tossing Mitsuo Kubo into the TV. Lastly, Adachi lacked tact, holding no degree of seriousness during his interactions with the Investigation Team and the protagonist's family--often going into topics in such a way that it usually drags down the mood or gets him beaten around by Dojima himself.
Magatsu-Izanagi also represents "Emptiness"; Adachi himself represents "Emptiness", distaste for the world at large, in Izanami's experiment to determine the value of humanity, which further shows Magatsu-Izanagi's connection to the Fool Arcana.
Using cheat codes in Persona 4, it is possible to have the protagonist equip Magatsu-Izanagi as a Persona, categorized as a Persona of the Fool Arcana. He has his own stats, a portrait, but no moves, so one must hack in skills for him to use. It is possible that Atlus intended for Magatsu-Izanagi to be available to the player normally, but did not finish implementing him. Alternatively, this may have simply been to test Magatsu-Izanagi without having to fight Adachi repeatedly.
As the negative form of Izanagi, he has different animations when attacking or casting spells. Izanagi looks more dignified, while Magatsu-Izanagi makes more arrogant and threatening movements. Also, unlike Izanagi, Magatsu-Izanagi wields his spear in an under-hand fashion.