Can we sort without permutations?
I don't have a definite answer to your question, but "braid sorting" seems a possible candidate. According to this wikipedia entry we can define it as follows. Let $X$ be a group, and let $H$ denote the set of tuples $(x_1,\ldots,x_n) \in X^n$ such that $x_1 \ldots x_n = 1_X$. If we let $G$ be the braid group $B_n$ generated by the moves $\sigma_i$,  we can define an action of $B_n$ over $H$ by:


  $\sigma_i(x_1,\ldots,x_n) = (x_1,\ldots,x_{i-1},x_{i+1},x_{i+1}^{-1} x_i x_{i+1},\ldots,x_n).$


That is to say, $\sigma_i$ combines the effect of a swap and a conjugation at positions $i$ and $i+1$. It might be possible to solve this problem optimally in polynomial time, which would  answer to your question.