Kings and Assassins
This one shouldn't be too much of a surprise to anyone who read the entirety of this article's introduction. Kings and Assassins is a game where one side plays the king, and the other plays the hidden sellswords who are trying to kill the king.
The asymmetric mechanics here open up the possibility of bluffing your opponent, sending them the wrong way in search of their target, then letting them fall into a trap. It's one thing knowing your opponent's moves, but it's another thing entirely figuring out who they're moving.
Click the links to download Kings and Assassins for iOS / AndroidiOS
You could really have added any multiplayer card game onto this list, but there's something about Summoner Wars that makes it stand out from the crowd. Sure, it's not Hearthstone, but it's got some brilliant ideas of its own that are well worth checking out.
The asymmetry here comes from the fact that you're building your own deck, based on a specific faction. That means that the cards you and your opponent bring into the arena are going to be very different. Finding a way to play against your foe's deck is the key to success.
Click the links to download Summoner Wars for iOS / Android
For some reason this one gets ignored quite a lot when people are talking about digital board games, but it's well worth checking out. It's all about the cold war, with players taking the side of the communists or the west.
Obviously both of these sides have their own way of playing, and that's where the imbalance comes from. Winning as one side is pretty different from winning with the other, although the goal of world domination / survival is still the same.
Click the links to download Twilight Struggle for iOS / Android
In this game set in the ancient world, players pick one of seven different civilizations to play as. That civilization must build one of the titular wonders if it wants to win. And there are different ways to go about creating those wonders.
The paths to victory might be quite similar in the game, but every faction has its own way of doing things. While the game still has standard victory conditions, it's how you and the other players try to get to them that the asymmetry comes from.
Click the links to download 7 Wonders for iOS / Android
Small World 2
Another game where the lack of symmetry comes from a choice the players make at the start of the game. The side and special move you choose will define how you're going to play throughout the rest of the experience.
That not only gives players plenty of scope for experimentation, it also means that the other people in the game are going to be left wondering about the path you're going to take towards victory. The game handles its inherent imbalance smartly though, and there's always enjoyment to be had here.
Click the links to download Small World 2 for iOS / Android