Here's the summary:

Initial Shuffling: Cards shuffled by rival's device $($Shuffle $1).$

Inserting Cards Back: Cards inserted back in same order.

Second Shuffling $($Shuffle $2)$: Rival shuffles again, revealing new order: J$,9,$ Q$,$ A$,$ K$,3,5,2,6,4,7,10,8.$

Reversing Order: Reversing Shuffle $2$ gives original order: Jack, $9,$ Queen, Ace, King, $3,5,2,6,4,7,10,8.$

Understanding the deterministic nature of shuffling and the reversal of the process allows for reconstructing the original order, neutralizing your rival's advantage.