Bray-Curtis is a general purpose multidimensional scaling technique that can be of use to analyze any multidimensional space. It simply creates axes in which to accomodate statistical units by using as poles the distance vectors with more negative correlation coefficient (that in practice means using as extremes the two A and B samples in which that maximizes the rule 'if sample i is very near to A it is expected to be very distant from B' and iterates the same procedure on the residuals of the previous step to generate subsequent axes. This is a very brilliant and simple common sense rule that works vry well.

Bray Curtis :)

erm, I think @allessandro giuliani is referring to polar ordination, which is indeed described in Bray and Curtis' seminal paper. The question here, though, is about defining resemblances among samples. Bray-Curtis similarity (or dissimilarity) is appropriate for situations where a zero implies absence. If you want to create a distance matrix using a single diversity measure, then use Euclidean distance, sure. if you want to combine a number of diversity measures, then normalize them (subtract mean, divide by standard deviation) and then calculate Euclidean distance. In terms of the 'test' you want to do ('metrics more similar upstream vs downstream sites') this actually sounds like a job for ANOSIM (which is a type of Mantel test).

As a matter of curiosity, some diversity metrics are logarithmic in nature (e.g. the Shannon-Weiner index). Then, would it be OK to ordinate samples using linear distances (like Euclidean)?

I agree with Paul Somerfield. In summary, for your case, I would normalize the data (if you use variables with different units) and then calculate resemblance based on euclidean distance. for the statistical test, ANOSIM or PERMANOVA