How do I change order/grouping/level of pandas MultiIndex columns?

Since levels indices are no more mandatory you can have even more simple way to achieve the level swapping of multi-index dataframe:

df = df.swaplevel(axis='columns')

Your columns are a MultiIndex. You need to reassign the DataFrame's columns with a new MultiIndex created from swapping levels of the existing one:

df.columns = df.columns.swaplevel(0, 1)
df.sort_index(axis=1, level=0, inplace=True)
>>> df

month   '1Jan'                 'Feb'                 'Mar'              
        weight  extent  rank  weight  extent  rank  weight  extent  rank
year                                                                    
2000      45.1  13.442    13    46.1   14.94    17    25.1   15.02    14
2001      85.0  13.380    12    16.0   14.81    15    49.0   15.14    17
2002      90.0  13.590    15    33.0   15.13    22    82.0   14.88    10
2003      47.0  13.640    17    34.0   14.83    16    78.0   15.27    22

You can then export to csv:

df.to_csv(filename)