Whenever we can interchange summation (perhaps due to Tonelli-Fubini), good things happen. Otherwise, one has to struggle evaluating double sums in just one way, because the alternative results in a divergent series. Having said that, I'm currently interested in the following:

Have you encountered in your own research or do you recall from someone's research paper "interesting" double sums that are convergent where reversing order of summation does not work? Please provide references.