vixter commented on my "dysspatialrotatia" post in the knitting community. She said that the way I was knitting was called "Eastern." I googled "eastern knitting" and found these:

http://www.modeknit.com/combined.html
http://www.modeknit.com/purl.html

They claim to show knit and purl stitches that are a combination of Eastern and Western. In the knit stitch, the needle goes in the back of the stitch from right to left (what I was doing), but the yarn goes around from left to right (I was doing right to left).

So I still don't know how the actual "eastern" knit or purl stitch work, but it's instructive to learn that other people do in fact knit in ways other than the standard method taught in the knitting books I looked at.