Yep, heard that advice. Chain wasn't too worn, though, as far as stretching, based on the stretch checking go-no-go tool I have. Cogs and chainrings look decent, too,

However, after some more digging here, I see that sometimes cross-chaining will lead to the behavior I described. The chain wasn't fully cross-chained (middle cogs with big ring) but once I switched to middle cogs with middle ring, I didn't see the chain jumping around when pedaling backwards. There's also a long discussion about whether pedaling backwards proves anything but I won't go there.