Answer by vysar for Swapping rooks in a 4x4 board

This is not a (new) answer to the original question, but I don't have enough reputation to comment. I tried to address the call for generalization using a similar technique as Jaap. Below the results for the board sizes that fit in my main memory. Unfortunately, 6 x 6 does not fit.

size    # configs  w   b=========================3 x 2         180  12  133 x 3        3360  16  173 x 4       69300  20  193 x 5     1513512  24  233 x 6    34306272  26  273 x 7   798145920  30  314 x 2         840  10  114 x 3       36960  14  154 x 4     1801800  18  194 x 5    93117024  22  234 x 6  4997280288  26  275 x 2        2520  10  115 x 3      200200  14  135 x 4    17635800  18  175 x 5  1647455040  22  216 x 2        5940  10  116 x 3      742560  14  136 x 4   102965940  18  177 x 2       12012  10  117 x 3     2170560  14  137 x 4   435134700  18  178 x 2       21840  10  118 x 3     5383840  14  138 x 4  1472562000  18  17

The last two columns give the minimal number of steps (ply) to the final position with either white (w) or black (b) to play.

The number of configurations is given by: $2 \cdot {n \cdot m \choose m} \cdot {n \cdot m - m \choose m}$, with $n$ the number of rows and $m$ the number of columns.