On the face of it the xor of two random bits should have the same distribution as the original bits unless the bits are related in some fashion. I guess the trick is determining whether there is any coupling between the pairs of bits that would be xor'd together.

determining whether there is any coupling between the pairs of bits that would be xor'd together

I read in that same text that in a good digest function, each bit of the input should affect approximately half of the bits of the output. (I think) the implication was that if this ideal is achieved, and the distribution of the affected bits for each bit of input is sufficiently unpredictable, then the digest is 'secure' meaning unpredictable. But the problem with many of the digests that have been shown to be weak, is that the ideal is not achieved, making certain patterns of outputs for given inputs discernable.

I read that to mean that there is coupling between the upper and lower halves of any digest. Good or bad?

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Coupling between halves is of less concern than if there is coupling between the bits that are xored together. My understanding of the MD5 algorithm suggests that coupling would be low and that xoring the top and bottom halves of an MD5 should retain the distribution characteristics of the non-combined digest (scaled to the smaller range of course). What's the application?

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