How do I easily flip odd and even pins between a Raspberry Pi and Pi hat?

My group built a custom hat for a Pi but when we integrated it we discovered a problem with our implementation.

Our even and odd pins were reversed. On the Pi, the headers are

------------
2  4  6  ...
1  3  5  ...
------------

But our board has the pins

------------
1  3  5  ...
2  4  6  ...
------------

Our 40-pin headers are surface-mounted on the bottom, so we cannot easily turn it around or put it underneath. We can connect a bunch of jumper wires and the circuit works by flipping the order of pins, but it’d be great if there was an easier workaround.

Is there some kind of thing we could get to do the reversal, or should we just do a revision on the boards?