Rearrange a PDF so that when the sheets are printed double-sided and folded down the middle, the pages read in the right order. This reordering is called imposition, and it is not something you can do by eye: on a folded booklet, page 1 has to share a sheet with the last page, page 2 with the second to last, and so on inwards.
That is what correct imposition looks like. The sequence only makes sense once the sheets are printed on both sides and folded - reading the imposed PDF on screen in page order is expected to look wrong.
A folded booklet always uses four pages per sheet, so the tool pads the document with blank pages to reach the next multiple of four. Adding your own blanks at the end gives you control over where they land.
Flip on the short edge for a booklet folded like a book. Flipping on the long edge prints alternate sides upside down, which is the usual cause of a booklet that comes out with every second page inverted.
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