PISO shift register with first input
If you need something different from what you designed into your board then you will have to change the board. Saying you cannot change the board is making an artificial restriction that will make it next to impossible to achieve the intended result without adding way more complication and cost than what you already have. 

Face the fact that sometimes mistakes are made and board spins become necessary. Although sometimes one can get lucky and adjust to keeping it the way it is and make the design work as is.