When a column on the grid is re-ordered there appears to be only the DisplayIndexMap object in the grid to keep track of the new order. This object is protected so I cannot get at it directly. Since I could not figure out how to determine the new order on the grid side I changed the column order of the datatable to match the grid's.
In the ColumnDisplayIndexChanged event of the grid I added this line of code, which takes care of the re-ordering:
This event will be triggered twice. Once for the column that was moved and again for the column that was shifted over for the moved column.
When I print the contents of the grid I use the datatable as the data source.