Nope, what I am suggesting that in the main model, you have the transposed matrix which you want to show. This is the one which operates on data. The next one on top of it merely transposes the transposed matrix and makes it as you want to see. It means making a few models. A model creation doesn't take too much time.
However, I think I am unable to express it clearly.
You can as well follow Chris advice and do the sorting in your model itself. At the row button click, get all the rows and their data and emit signals like columnsAboutToBeMoved(), columnsMoved, etc.