https://discuss.asciidoctor.org/Get-page-count-into-the-document-tp8296p8302.html
I'll be honest, I'm really confused about what you are trying to accomplish. Now that I see your example, it's not the same thing as facing pages.
Since pages are continuous, there's no real concept of a left page per se. It all depends on which page you start counting on. Any page can be a left page if you just look at it and say the next page is the right page. So I don't understand why that isn't what you already have.
> Can you tell me if it is possible to have a "== Introduction" chapter,
where this is part of the TOC, but not shown on the page where the
chapter starts? That way I can define a "chapter page" that contains
only the name of the chapter and a '<<<' at the end
No, that's not possible.
I'll be honest, it kind of feels like you are using Asciidoctor PDF for something it wasn't intended to do. Asciidoctor PDF was not designed to make slides. If that's what you want, you may want to consider using Asciidoctor Reveal.js instead (which has a PDF export option).
Best Regards,
-Dan