TITLE: Rearrangements of absolutely convergent series
QUESTION [8 upvotes]: Accounts of absolute and conditional convergence always say that
$$
\sum_{n=0}^\infty a_n = \sum_{n=0}^\infty a_{\sigma(n)}
$$
if the series converges absolutely, if $\sigma$ is any bijection from the set of indices to itself.
Fubini's theorem tells us that
$$
\sum_{n=0}^\infty\sum_{m=0}^\infty a_{n,m} = \sum_{m=0}^\infty\sum_{n=0}^\infty a_{n,m}
$$
if we have absolute convergence.  (Tonelli's theorem tells us that these are equal if all terms are nonnegative, regardless of whether the series converges or not.)
But in this answer, I explained how to rearrange
$$
\sum_{n=0}^\infty\sum_{m=0}^\infty a_{n,m}
$$
into
$$
\sum_{n=0}^\infty \sum_{k=0}^n a_{k,n-k}.
$$
What known theorems apply to this case, and where is this case mentioned in the literature?
How exotic does the class of rearrangements of series get?  What other kinds of rearrangements don't fit instantly into one of the cases covered by the two categories above that are covered by standard results?
(I might post my own answer to this.)

REPLY [2 votes]: It seems to me that you are alluding to the theory of unordered summation, which is discussed (for instance) in $\S$ 14.2.3 of these notes.  
In particular it is discussed that unordered summation is equivalent to absolute convergence, and the application to double series is given as an exercise.
(In my opinion calling this phenomenon "Fubini's Theorem" is a bit fancier than necessary.)