Interestingly (to me anyway) I've had a similar problem that I've been working on for a couple of years. I'd describe the specifics differently, but it still comes down to my heelside turns being handled differently from my toesides.
On any kind of steeps, I'll find that I turn more to heelside and tend to skid or skitter on my tail a bit. This has been consistent enough to be a real problem on steeps. Today on Seymour (also after having read this thread), I paid special attention to this. This is what I came up with, and when I corrected it I found my steeps handling improved.
First, I do simply tend to turn more away from the fall line when going heelside. This immediately means more pressure on the edges. I'm not sure of the motivation -- less comfortable with going downhill when I can see it?
Second, I don't normally do the flexion/extension thing, at least not as much as I should. This means that once I'm on the heel edge and essentially braking, it doesn't ease up.
Third, although I have no problem committing to the next turn when I'm on toeside, I find a distinct reluctance to lean down the hill with my upper body when I'm facing downhill.
Today I made a point of not turning to go completely across the fall line heelside; I worked on getting the flexion/extension going on both sides; and most importantly I tried to commit to initiating the heelside turn.
Made a big difference.
I hate the parts between winter