Well, I clearly got this thread off on the wrong foot, and I'm sorry. Sometimes I get into these strangely flippant moods but I hope people can see past that. Can we start over?
Cage's observation about Twombly, which though it stuck with me may not have done him proper justice, was that his guileless and uninhibited 'scribblings' encouraged Cage to appreciate those textures in the world that came about with no artistic intent at all, such as the weathering patterns on a New York sidewalk. This resonated strongly with me, so strongly in fact that when I encountered a Twombly work that seemed too "made" I felt compelled to relegate it to some back bench rather than really look at it.

I suppose the trouble with this image for my sensibilities is that the iconography of it is really too reminiscent of graffiti, as if Twombly had all too consciously embraced the characterizations of his work formulated by others, i.e., that his work was reminiscent of the work of graffiti artists. So my flippancy might be traced to reviews of this slightly cringeworthy ilk:
http://www.guardian.co.uk/artanddesign/jonathanjonesblog/2008/jun/03/cytwomblytheonlygraffitia -- not that I'm seeking an excuse. More like a dubious reason to justify why I haven't given the above image a proper reading. I have never seen it in person, either, by the way, so that's an additional ding against me as a commentator.
I'm far more prepared to see Twombly in the tradition of Pollock, which is still an extremely facile connection. It doesn't feel right to me, and this time I could perhaps venture a guess as to why. Pollock's paintings really were 'composed' with a sense of rhythm and contrast and to some degree at least a sense of pressure to 'fill the canvas' which even as long ago as Cézanne was a dictum artists were prepared to violate. Of course, Pollock's paintings would have lost their intensity if they hadn't just seemed to overwhelm the edges and threatened to burst beyond them. Anyhow, Twombly's intensity was of a very different variety, and in his best work it seemed as if that intensity was an attribute of the lines rather than of a sense (however intuitive and 'guileless') of artistic 'vision'.
You can see that my thoughts are all over the place, and I'm not sure I have made things any better with this post except to own up to my fuddle. As I give the above painting a second look, less encumbered by an externally imposed bias against 'made'-ness, I see a kind of foreground-middleground ambiguity that appeals to me very much indeed. The background is, of course, clearly identifiable as the dark blue, slightly green, texture in the, well, background. However, it seems that the conventional idea that brighter colors are more 'forward' is frustrated in a very specific sort of way. The more muted loops (which are also tendentially higher and larger) do not always acquiesce to being middle ground do they?
Talk to me about Cy Twombly. May he rest in peace.