For the purpose of recording an answer rather than just a pile of links: Michael Hardy requires that

>if either limit exists then so does the other and in that case then they are equal?

Let's call this set $G$.

Levi answered a slightly different question, namely characterizing the permutations for which

>if the left hand limit exists then so does the right and in that case then they are equal?

I'll call that $P$. Clearly, $G = P \cap P^{-1}$.

<b>Theorem</b> A permutation $f$ is in $P$ if and only if there is a constant $M$ such that, for any $N$, the set $f([1,N])$ can be written as $\bigcup_{i=1}^M [a_i, b_i]$, where $[a,b] = \{ a,a+1, \ldots, b \}$.

[Levi][1] provided a different characterization than this one; [Agnew][2] provided this characterization; I learned about both from [Schaefer][3] who points out that they are fairly directly equivalent.

[Pleasants][4] shows that $P$ is not closed under inversion. I haven't (in an one hour skim) found any papers which give a simpler characterization of $P \cap P^{-1}$ than the defining formula.

<b>Remark</b> I would find it nicer to restate Levi/Agnew's characterization as follows: For $S \subseteq \mathbb{N}$, define the blockiness of $S$ to be the least integer $\beta(S)$ such that $S$ can be written as $\bigcup_{i=1}^{\beta(S)} [a_i, b_i]$. (We could have $\beta(S) = \infty$.) Then $f$ is in $P$ if and only if there is a constant $M$ such that $\beta(f(S)) \leq M \beta(S)$. This makes it more obvious that $P$ is closed under composition.



  [1]: http://projecteuclid.org/euclid.dmj/1077473866
  [2]: http://www.ams.org/journals/proc/1955-006-04/S0002-9939-1955-0071559-4/home.html
  [3]: http://www.math.ust.hk/~mamyan/research/UROP/schaefer.pdf
  [4]: http://dx.doi.org/10.1112/jlms/s2-15.1.134