I can't believe it! I figured it out. My brain is near the melting point, but I did it.
I didn't reverse my work entirely, just one stitch per needle per set. If that makes sense.
So instead of:
Knit off, purl on; purl off, knit on.
knit on(front) and purl on (back) to start
Purl slip off, purl keep on (front)
Knit slip off, knit keep on (back)
Knit off, knit on (front)
Purl off, purl on. (back)
I don't know if that makes any sense the way it is written, but it works.