The quick answer is to turn OFF (disable) the Mirror Pages option on the Layout tab:

This will fix the pages swapping sides, but you will probably need to manually override the header settings to get the appropriate script(s) on each left/right (or inner/outer) combination. (Let me know if you need any assistance with that).
[My guess is that in future we need to rethink what the mirror setting should do, and possibly split it into 2 controls that control pages and header information separately.]