How to order column groups by a cell's value

I have a spreadsheet of multiple datasets in columns. I checked some documentation here, https://support.google.com/docs/answer/3093150, but the SORT() function would order my columns on a column-by-column basis when I want to keep pairs of columns together. The first row has axis labels and the second has dates I want to sort by.

For example, this is what I have.

A      B      C      D      E      Fdate2  null   date3  null   date1  null1.000  .2353  1.000  .3235  1.000  .28232.000  .4593  2.000  .3852  2.000  .39243.000  .6372  3.000  .7238  3.000  .6932

But I want

E      F      A      B      C      Ddate1  null   date2  null   date3  null1.000  .2823  1.000  .2353  1.000  .32352.000  .3924  2.000  .4593  2.000  .38523.000  .6932  3.000  .6372  3.000  .7238

where date_i are now sorted.