How to make a doublesided longtable with a doublesided header?

I'm currently struggeling to extend a onesided longtable with fixed column widths to a double-sided layout (where the position of the columns changes depending on even/odd pages). My current approach would be to change the table to a onecolumn layout and switch the layout of the rows depending on left/right pages, but keeping the same row heights - thus avoiding instable solutions. This will most likely work - but I also need different table headers on the pages.
But longtable as is stores the table head in a box, so the headers are the same on every page. Is there a way to get different headers on even and odd pages with longtable?