First off, an interesting aside. Someone else pondering your question led indirectly to the creation of the entire fantasy genre:
"I first tried to write a story when I was about seven. It was about a dragon. I remember nothing about it except a philological fact. My mother said nothing about the dragon, but pointed out that one could not say 'a green great dragon', but had to say 'a great green dragon'. I wondered why, and still do. The fact that I remember this is possibly significant, as I do not think I ever tried to write a story again for many years, and was taken up with language."
From "Letters of J.R.R. Tolkien".
To answer your question:
Are there instances in which adjectives are non-commutative, so that permuting the order of the adjectives changes the literal meaning of what's being said?
Yes, lots. For example, consider the mathematician's definition of "diameter". Suppose you have a shape -- any shape, in any number of dimensions -- and you want to find its diameter. How can we define diameter on an arbitrary shape?
It's actually straightforward. Consider every possible pair of points on the object, and consider every possible "path" between those two points. For each pair of points, find the shortest of those paths. The length of that path is the shortest distance between those two points. Now find the pair of points that has the longest shortest path between them. The length of that path is the diameter.
Clearly the "longest shortest path" is very different than the "shortest longest path".
See also my article about finding the longest shortest shortest path between poppyseeds on a bagel, if this subject interests you.