OK, I think I get it. Garter stitch often looks a lot like reverse st st.
To be honest, though, if you're getting true garter on both the inside and outside with the dpns without purling, I haven't a clue what's going on. If your v's are turning up on the inside then you need to spin the work around so your needles are close to you. You'll have to take out the stitches you've done, but all of us have had to do that.
You switched correctly, but I think you're knitting on the inside of the hat now, which is a matter of where you're holding the needles.
"Trust the pattern!"
Time flies like an arrow, but fruit flies like a banana.
Hop aboard while we follow the cloud-covered van of knitting harmony, man!! Woooo!! --Hedgehog
IngridKH on Ravelry