How popular would this reverse type battle be?

[Paper Mario]

Type matchups are reversed. Unlike an inverse battle, immunities are reversed too and types that resist or are weak to themselves remain as so.

So under normal battle conditions, Bug is strong against Dark and Psychic (while both deal normal damage to Bug), Ghost is immune to (and deals normal damage to) Fighting while Fairy is immune to (and is strong against) Dragon. Bug would instead be weak to (and deals normal damage to) Dark and Psychic, Fighting would be immune (while dealing normal damage) to Ghost and Dragon would be immune to while also being strong against Fairy.

[Lord_Brand]

I'm not a big fan of chart-wide inversions like this, barring maybe some kind of Room move or other in-game explanation for physics being warped.

[Shanty Pete's 1st Mate]

Honestly, I think this format would be substantially more confusing than Inverse Battles as they exist. But if they were to happen?

- Offensive-Steel becomes the worst type in the game, with 9 types resisting (Normal, Flying, Fairy, Steel, Rock, Ice, Bug, Grass, Psychic) and 1 type immune (Poison), but only 3 types weak-to (Fire, Fighting, Ground).

- Defensive-Normal becomes elite, retaining its Ghost immunity and trading its Fighting weakness for Steel and Rock resistances.

- Offensive-Grass would be great, too, being super-effective against 5 types (Poison, Bug, Ice, Fire, Flying) and not-very-effective against 4 types (Grass, Water, Ground, Electric). Huh, actually more balanced than I thought.

- Defensive-Electric is now immune to Ground, but weak to Flying and Water. Might be a worthy trade, honestly.

[Emerson]

Considering Inverse Battles, while super cool, were already very ignored, it's not something I could see many people being interested in relatively.
That said, Super Effective/NVE identifiers on the UI were added after Inverse Battles were removed, so it may be a more accessible battle style now.

But ultimately. I have to agree that it'd be better off as more of a main story gimmick/post game challenge or move effect rather than a whole new battle style. Extra battle styles have sadly kind of been dead in the water without even getting off the ground, Triple/Rotation barely ever had a scene and Inverse even less so (triple/rotation was already kind of a bad idea anyways).

I also honestly think a Room move might be a bit too much as well. It's significantly more impactful and difficult to work around than TR. Ideally, imo, you'd make it a status like Soak, Trick-or-Treat, etc, where the afflicted Pokemon's type matchups are reversed after being hit. This would let you use it both offensively and defensively by targeting your ally while being much more manageable for both parties. I think you'd even want it to be a single turn thing, although that'd obviously make it a Doubles only strategy (which is fine).