I was utterly stuck on the second level until suddenly I found myself carrying the top 2/3 of the pillar with my tail. I decided to restart the level so I could do it deliberately...and now, even knowing that that's the goal, I have absolutely zero idea how I achieved it.
I'm as much of a fan of "figure out the mechanics" as the next thinky player, but I feel like a little explanation would go a long way here.