1. Can I trade cards between two opponent's racks?

No, a Trade has to go from your rack of cards to 1 opponent's rack of cards.

2. Do the cards flip when I do a Trade?

No, you will see the same side of the cards before and after a trade.  Another way to think about this is, the outside of the card before you trade becomes the inside of the card after the trade.

3. Can I Swap the position of cards in my opponent's rack?

Yes, a Swap or Flip can be done to cards in any rack.

Rules Variants:

1. Team play: If you have 4 players, then you can play "Team Patchwork".  In this variant, the 2 players sitting opposite each other are a team (kind of like in Bridge) and the players on a team can work together to setup larger sets, particularly on the backs of their opponents' racks.  At the end of the game the teams count up their combined number of cards and the higher total wins.

This variant changes the game play and some strategy significantly.  It changes when you might steal cards from your partner and allows you and your partner to setup a scoring on the opponent's hand that sits between you.  

But table-talk isn't allowed - if you are setting your partner up - they have to figure it out on their own.