An important point on the Husserl episode that I was trying to get across was his notion that "intentionality" as he uses it doesn't just mean that all conscious acts have a target, i.e. something you're conscious of, but that this content is not itself something subjective. When we grasp something in consciousness, we're not just contemplating our own sensations (as Schopenhauer describes our inner sense checking out and making sense of the data fed in by our outer sense). Rather, consciousness is a connection between us and something objective: you and I in general can experience the same objects, whether they be physical objects or even the notion "Santa Claus." If you and I think about that, we're thinking about the same thing, which of course raises the question of what this thing is. Frege considers this "sense" that we both contemplate to be an objective entity that we have to admit into our ontology: we can't take intentionality seriously and be materialists.
In reading Martin Heidegger's The Basic Problems of Phenomenology,I found a discussion of this around p. 62:
Intentionality is said to be a character of experiences. Experiences belong to the subject's sphere. What is more natural and more logical than to infer that, consequently, that toward which immanent experiences are directed must itself be subjective? But however natural and logical this inference may seem and however critical and cautious this characterization of intentional experiences and of that toward which they direct themselves may be, it is after all a theory, in which we close our eyes to the phenomena and do not give an account of them themselves.
...To say that I am... oriented toward sensations is all just pure theory... Perception is directed towards the extant being itself... This holds also when I am involved in a perceptual illusion. If in the dark I mistake a tree for a man, it would be wrong to say that this perception is directed toward a tree but takes it to be a man, that the human being is a mere representation and, consequently, in this illusion I am directed toward a representation. On the contrary, the sense of the illusion is precisely that in taking the tree for a man I am apprehending what I perceive and what I believe I am perceiving as something extant. In this perceptual illusion the man himself is given to me and not, say, a representation of the man.
...The question as to how subjective intentional experiences can on their part relate to something objectively present is put completely the wrong way. I cannot and must not ask how the inner intentional experience arrives at an outside... because intentional comportment itself as such orients itself toward the extant. I do not first need to ask how the immanent intentional experience acquires transcendent validity; rather, what has to be seen is that it is precisely intentionality and nothing else in which transcendence consists.
Now, there's a question re. whether I should really be attributing this conception of intentionality to Husserl; Hubert Dreyfus (whose Heidegger lectures I've been listening to), thinks that Husserl's view is more like Descartes's here that Heidegger is attacking, but I'm not so much interested in the interpretive question if the idea itself is the one we should be shooting for.