How much do self organising maps suffer from local minima problems? I assume that the original orientation of the grid can have major orientation on the resulting shape of the grid, but does this have large impacts on the fit (e.g. mean square difference from data to assigned grid points), or are the differences fairly small?
If your data is high-dimensional, there will be many local minima, and the optimization problem is almost combinatorial. Then it is likely, but not guaranteed, that you will have a large number of optima that are almost as good as the global optimum—which you will never find.
That is, there is no simple answer. It definitely is prone to local minima. Try different initializations and choose the result with the smallest MSE.