Inversion is a phenomenon where an entire city is erased.
Inversions are said to be rare and a sign of major instability for the areas in their vicinities. They entail the barrier between a city's Underground and Realground being destroyed, resulting in the two planes "cancelling each other out." This is caused by special Noise created through Dissonance, with two examples being the Plague Noise and the Soul Pulvis.
In the aftermath of an Inversion, all of a city's erased residents are reduced to disembodied thoughts with no physical presence. In addition, the city itself is wiped from the memories of everyone in the RG (though certain individuals who had experienced the UG previously may be able to preserve their memories).
When Shinjuku was erased, a red Reaper skull appeared in the sky, pulsing out and disintegrating government skyscrapers. Aside from the Reapers of Shinjuku, the only survivor was Tsugumi Matsunae, herself a former Reaper whose Soul was sealed inside Mr. Mew.
During the Inversion of Shibuya (before it was undone by the Replay power), the symbol appearing in the sky was instead that of Phoenix Cantus. All structures in the city crumbled into pieces that floated upwards as the area turned red. All Players and Reapers seen were erased when they were swarmed by the Soul Pulvis.
Inversions first appear in the series in The World Ends with You: Final Remix's exclusive epilogue chapter, A New Day. It begins with Neku having a vision of the Shinjuku Inversion. He considers them flashbacks, but when he sees Rhyme erased by the Swing Shark he realizes they're premonitions. Accordingly, one of the final scenes of A New Day is the Shinjuku Inversion, and Hanekoma and Joshua discussing it. Both see it as concerning to happen so close to Shibuya, but while Hanekoma is curious about the repercussions, Joshua is disinterested.
In the Secret Reports, Hanekoma writes that the Shinjuku Inversion happened before Beat and Neku disappeared from the RG. He notes that the creation of the Dissonance Tapir and the resurrection of Sho Minamimoto happened around the same time.
After the Inversion of Shinjuku, its Reapers moved to Shibuya.
It is eventually revealed that Shiba Miyakaze caused the Shinjuku Inversion under the manipulations of Tanzo Kubo, who himself had been recruited to do so by Hazuki Mikagi. Kubo then plotted to bring about Shibuya's Inversion as well, again using Shiba as his pawn. As a backup plan, Kubo grants Rindo Kanade the power of Replay. Continuous use of it would produce the Soul Pulvis from the Souls of undone timelines, which could be utilized if Shiba was thwarted or abandoned his efforts.
In Week 3, Day 1, Shiba reveals that his Plague Noise are capable of eroding the barrier between planes and bringing about the Inversion. After Shiba's defeat on Week 3, Day 7, Kubo unleashes the Soul Pulvis from Rindo's Player Pin, triggering the Inversion. After all the Wicked Twisters save himself are erased, Rindo uses Replay again to try and avert this. However, this only makes the Noise stronger; an attempt to fight them proves fruitless and all of Rindo's allies are erased again. The Inversion is halted, however, by the intervention of Hazuki, who exorcises Kubo and erases his Noise.
After Rindo rewinds time again to save his friends, he and a large amount of allies execute Operation: Awakening, a plan to weaponize Shibuya's living Souls against the Soul Pulvis. The plan succeeds, causing the Noise to coalesce into a single entity--Phoenix Cantus. After the Wicked Twisters succeed in erasing it, the Inversion is fully averted.