The first clue about the theme, so to speak, of this sutra is to understand it is one of the Prajnaparamita -- perfection of wisdom -- Sutras. These sutras are associated with the second turning of the dharma wheel. The significance of the second turning is the development of the doctrine of sunyata and the ideal of the bodhisattva who brings all beings to enlightenment.
The sutra represents an important milestone in the development of Mahayana. In the first turning teachings of Theravada, much emphasis was placed on individual enlightenment. But the Diamond takes us away from that --
"... all living beings will eventually be led by me to the final Nirvana, the final ending of the cycle of birth and death. And when this unfathomable, infinite number of living beings have all been liberated, in truth not even a single being has actually been liberated.
"Why Subhuti? Because if a bodhisattva still clings to the illusions of form or phenomena such as an ego, a personality, a self, a separate person, or a universal self existing eternally, then that person is not a bodhisattva."
I don't want to belittle the importance of the doctrine of impermanence, but impermanence was expounded by the historical Buddha in the first turning teachings, and the Diamond is opening a door to something beyond that. It would be a shame to miss it.
The several English translations of the Diamond are of varying quality. Many of the translators have attempted to make sense of it and, in doing so, have utterly scrambled what it's saying. (This translation is an example. The translator was trying to be helpful, but in attempting to render something intellectually graspable he erased the deeper meaning.) But in the more accurate translations, something you see over and over is a conversation like this:
The Buddha: So, Subhuti, is it possible to speak of A?
Subhuti: No, there is no A to speak of. Therefore, we call it A.
Now, this doesn't just happen once. It happens over and over (assuming the translator knew his business). For example, these are snips from Red Pine's translation--
(Chapter 30): "Bhagavan, if a universe existed, attachment to an entity would exist. But whenever the Tathagata speaks of attachment to an entity, the Tathagata speaks of it as no attachment. Thus is it called 'attachment to an entity.'"
(Chapter 31): "Bhagavan, when the Tathagata speaks of a view of a self, the Tathagtata speaks of it as no view. Thus is it called a 'view of a self.'"
These are a couple of random examples I picked mostly because they are brief. But as you read the sutra (if the translation is accurate), from Chapter 3 on you run into this over and over again. If you aren't seeing it in whatever version you are reading, find another one.
To fully appreciate what is being said in these little snips you need to see the larger context. My point is that to see what the sutra is pointing to, here is where the rubber meets the road, so to speak. It makes no intellectual sense, so people paddle by these parts of the sutra until they find firm ground on the "bubble in a stream" verse. And then they think, oh! This is about impermanence! But this is making a huge mistake, because the parts that don't make intellectual sense are critical to perceiving the Diamond.
How to interpret these "A is not A, therefore we call it A" teachings? I hesitate to presume to explain it, but I partly agree with this religious studies professor:
The text challenges the common belief that inside each and every one of us is an immovable core, or soul--in favor of a more fluid and relational view of existence. Negative, or seemingly paradoxical statements by the Buddha abound in the text, such as "The very Perfection of Insight which the Buddha has preached is itself perfection-less."
Professor Harrison elaborated, "I think the Diamond Sutra is undermining our perception that there are essential properties in the objects of our experience.
"For example, people assume that they have "selves." If that is the case then change would be impossible or it would be illusory." said Harrison. "You would indeed be the same person that you were yesterday. This would be a horrifying thing. If souls or "selves" did not change, then you would be stuck in the same place and be as you were when you were, say, two [years old], which if you think about it, is ridiculous."
That's a lot closer to the deeper meaning than saying the sutra is about impermanence. But I'm not sure I agree with the professor's interpretation of the "A is not A" statements, so I'll turn to Thich Nhat Hanh about that. This is from his book The Diamond That Cuts Through Illusion:
"When we perceive things, we generally use the sword of conceptualization to cut reality into pieces, saying, 'This piece is A, and A cannot be B, C, or D.' But when A is looked at in light of dependent co-arising, we see that A is comprised of B, C, D, and everything else in the universe. 'A' can never exist by itself alone. When we look deeply into A, we see B, C, D, and so on. Once we understand that A is not just A, we understand the true nature of A and are qualified to say "A is A," or "A is not A." But until then, the A we see is just an illusion of the true A."
Zen teacher Zoketsu Norman Fischer was not specifically addressing the Diamond Sutra here, but it seems to relate --
In Buddhist thought the concept "emptiness" refers to deconstructed reality. The more closely you look at something the more you see that it is not there in any substantial way, it couldn't be. In the end everything is just a designation: things have a kind of reality in their being named and conceptualized, but otherwise they actually aren't present. Not to understand that our designations are designations, that they do not refer to anything in particular, is to mistake emptiness. ["A Few Words About Emptiness," PDF]
This is a very crude attempt to explain a very deep and subtle sutra, and I don't intend to present it as the ultimate wisdom about the Diamond. It's more like trying to shove us all in the right direction.