How can I view PDF at two-pages and in the right-to-left order? How can I view PDF at two-pages and in the right-to-left order?

Acrobat can do this but Preview is more responsive.
Preview cannot do this.

You can manually rearrange each two-page layout in the sidebar, but this is time consuming.

If you don't care about the exact layout, you can add a blank page to the PDF in Preview (Edit > Insert > Blank Page) before the first page. This will shift the right page into left spot of the next two-page spread. This does break right-to-left formats, like manga.

There may be other apps that can accomplish this for you. Like you said, you can do it with Adobe Acrobat, and this is likely how you'll have to do it.