The maximum value is $4n^3$.

1. First consider the maximum value of the sum of horizontal distances: project everyone onto the same column, and connect the two people of each couple with a line segment. Consider two possibilities:
- There exist two couples whose connecting segments do not overlap: then take one person from each of the two couples and swap their positions, and the sum of horizontal distances becomes larger.
- The connecting segments of any two couples overlap: without loss of generality, assume that in every couple the man is to the left of the woman. We find that all the men must lie within the left $n$ columns, otherwise:
* If there is a couple with both the man and the woman in the right $n$ columns, then the right $n$ columns have only $2n^2 - 2$ seats remaining, but there are still another $2n^2 - 1$ couples that need to be seated.
* Therefore there must be a couple both of whom are in the left $n$ columns. However, a couple both of whom are in the left $n$ columns cannot have their connecting segment overlap with that of a couple both of whom are in the right $n$ columns, a contradiction!

In other words, the man of every couple must belong to the left $n$ columns, and the woman of every couple must belong to the right $n$ columns. It is easy to compute that the sum of horizontal distances in this case must be $2n^3$.

2. Similarly, the maximum value of the sum of vertical distances is also $2n^3$, so the total distance is at most $4n^3$.

3. Finally, we show that there is a way to achieve $4n^3$. Consider dividing the couples into two groups $A, B$, each with $n^2$ couples, and dividing the seats into four quadrants (each quadrant being an $n \times n$ grid). Let the first quadrant be seated entirely with $A$ men, the second quadrant with $B$ women, the third quadrant with $A$ women, and the fourth quadrant with $B$ men. Then one can verify that this construction achieves the maximum value estimated above, which completes the proof.