A5 Printing on A4 with custom flips and unique layout / flow direction

This looks like a specialized booklet output that you could assemble and then staple in the middle.

That usually requires either a printer with that function built-in (we have these at work) or a specialized "Imposition" program that will send a document like this to a printer that does not have this feature built in.

This is referred to as "imposition." Since your printer does not support imposition you need to get an application that will impose this particular layout for you and then send it to your printer. Sorry, but I have no idea what form such an app would take, but I do know that such add-ons are available for the likes Adobe InDesign or Quark Express.

Taking a PDF to a printing service provider would work. Office Supply stores often also offer these services.