Purgatorio

Purgatorio is the name of the realm that is parallel to the Human World in the Trinity of Realities. Though not technically part of the Trinity itself, it is caught in between the realms and is closest to the human concept of purgatory or limbo, as its name implies. As Antonio Redgrave explains in his journal entry regarding Purgatorio, the realm exists as a meeting ground that was used by the Umbra Witches and Lumen Sages to contact their heavenly and infernal masters or to battle without interfering in the affairs of the human realm.

Purgatorio, Center Of The Trinity
"One of the unique aspects of the religion in this town is the people's belief that the world is actually made up of three unique realms layered on top of each other – Human World, Inferno, and Paradiso – the Trinity of Realities.

''However, the wavelengths of the Spirit Energy in each of these realities is different, and inhabitants of each are unable to interfere with the other's affairs, despite the fact that the worlds are layered. It's said that Lumen Sages and Umbra Witches had the unique ability to pass into a realm known as Purgatorio, and travel amongst the Trinity. It is believed that the witches and sages each made contracts with the masters of the light and dark worlds, and can use special abilities granted to them by their masters.''

''Underlying all of this, though, is the presence of the Trinity of Realities. Even during the Witch Hunts, it seems those suspected of being witches were lured into Purgatorio, an act the people called being "spirited away," and their presence in the no-man's land was used as justification for further acts of persecution.''

To further explain Purgatorio, one must understand that those within the realm are unable to see or touch those in the human world, nor are humans in their own world able to intervene in the affairs of Purgatorio.

''Moreover, what we identify as material objects may have different appearances in the different realities. Yet, when these items are destroyed in one world, they will be destroyed in all worlds. This may explain why many place religious interpretations upon the phenomenon of unexplained destruction that occurs during their everyday lives".'' - Antonio Redgrave

Bayonetta & Bayonetta 2
In both games, Bayonetta travels into Purgatorio whenever she must conduct her battles with angels and demons in her way without causing interference in the human realm. When in Purgatorio, she becomes invisible to the other people in the Human World, while the humans are transparent, and can go about her business unseen. However, destroying an item or physical object when in Purgatorio will also destroy it in the other realities as well and she can still be seen in reflections in mirrors or windows.

Both Bayonetta and Jeanne have the ability to see beings in Purgatorio naturally as part of their status as Umbra Witches due to their connection with the true nature of the universe. For normal humans, such as Luka, they must use special eyewear that has been magically altered to allow vision into Purgatorio.

Trivia

 * In The Divine Comedy, Purgatory is a place where one goes when they lived a sinful life, but died in grace. It acts as an intermediating realm between heaven and hell.
 * The Infernal Communicator accessory states that it summons "Little Devils" from Purgatorio, meaning that some creatures inhabit the realm naturally.
 * In Hideki Kamiya's developer commentary, he notes that Bayonetta can be reflected by mirrors, even while in Purgatorio. This is a partial reference to Devil May Cry, where mirrors were portals to the demon world.