vibration (felt in sterring wheel) under load

My new-to-me 2001 Golf TDI has a strong vibration (I just noticed it on the drive home from buying it, and I haven't driven it since) in the steering wheel, worst case perhaps a span of 1/2" at the rim of the wheel. It gets worse with speed; I probably noticed it the first time around 50mph, and at 60-70, it's very pronounced.

It's there when accelerating, I haven't checked when coasting in neutral, but when I leave it in (5th) gear adn coast, it goes away.
What could that be? Something (ball joints?) loose in the steering, so that the tires can point outward or inward when pulling (and they jump back in or out to their track width, producing the vibration), while keeping them straight when the car is pushing the engine?
Is there any way this could be caused by out-of-balance tire alone? (What about rear tires?)

My vote is that it's simply an out of balance and/ or cupped tire. (Especially if it has the Goodyear Eagle LS).

I had a similar vibration in my car when it was new. It did turn out to be an out-of-balance tire. Try rotating the tires front to back and see if there is any change. Or, just get the tires balanced, as they probably need it, anyway.

Check for cupped tires...tread may look fine on first inspection but if you look closely the tread may have worn unevenley. Happened to me on all my tires after they are 2/3rds worn.
Byron

(Correction: it's only +/-1/8" of movement in the steering wheel. That's just already enough to feel massive. And it does also go away when coasting in Neutral; even more so than when coastign in gear)

So tires can indeed cause vibration that is comes and goes as dewcribeed above? How odd.
Thanks - I will try moving tires around ASAP.
Given that the vibration is in the steering wheel - does that mean it has to be caused by the front tires?
I do have another whole set of wheels with tires that are fairly worn, but perhaps they are differently-worn enough to show a difference.

Yes, the shimmy I felt in my steering wheel was almost exactly as you describe -- I could feel it only under acceleration or engine load, like climbing a hill. On level ground at constant speed or coasting, it went away. I thought I had a bad CV joint or something.

Anyway, another forum member suggested I rotate the tires, and sure enough, the shimmy went away.

I then took the wheels to a place with a Hunter GSP 9700 wheel balancer, and they found that one of the tires that had been on front was WAY off, and needed remounting. After that, I've had no problems at all.