There is a genuine asymmetry between our access to ourselves and out inner life on the one hand, and our access to the external world on the other. This asymmetric access had two aspects: in certain fundamental respects, we know ourselves better than we know others, and we know ourselves better than others know us. In retrospect, the discovery of asymmetric accesss is not all that surprising. Consider: what am I visualising right now? The correct answer is: a three-headed kangaroo. But how is it I know the correct answer when you could not?That last isn't the question, though. The question is: in what sense is this asymmetric? Or, since the answer is 'in no sense', the more pointed question is: how could anybody genuinely think there's any genuine asymmetry here?
Put it this way: the world doesn't know us (the nature of our thoughts about kangaroos); but there's nothing assymetric about this because we don't know the world either. Or more precisely: the world knows something about us, but not everything; and this exactly ('symmetrically') describes our situation with respect to the world ... we know something about it, but not everything.
Of course, if we had a perfectly comprehensive and transparent knowledge of the cosmos, some asymmetry might creep in. But quantum physics and chaos suggest such knowledge isn't in the grain of things. Or to put it another way: if we had a perfectly comprehensive and transparent knowledge of the cosmos, then the state of affairs would obtain in which the cosmos (of which we are a part) had become perfectly transparent, and our thoughts about many-headed kangaroos would be precisely as knowable as everything else.
There's something very profound in this, I think.