Actually, I think I found the own answer on another forum (from Artwork Abode):
1.Open up your pages panel.
2.In the pages panel menu you need to turn off Allow Document Pages to Shuffle.And now the document pages will not longer shuffle.
3.Now use the pages panel to pull the pages apart.You need to select just one of the pages on the spread.
4.Click and drag it out to the right, drag it until you see black vertical bar showed up.That means that you can let go of it, and the pages will be split into two.
You only need to do this in cases where there is content that needs to touch the spine without crossing over to the opposite page in the spread. If there is nothing in the inside margin area (where inside bleed comes from) you can simply export normally.
You did it wrong.
If a document is facing pages because it has to be printed on both sides it has to remain facing pages.
You need only to export the PDF as single pages.
I've got a book in spreads and need to supply individual pages (with bleed on all sides) to the printer.
Is the book's binding going to be Wire-O? Otherwise leave it as spreads and include an inside bleed when you export.