If you really want an unsymmetrical position, don't play
1...e5. White has many move order options to steer the game into a relatively quiet position in double king pawn openings.
There are a few things that you can consider in order to keep the game interesting. First and foremost, if you must win at all costs, I would recommend playing
4...Be7 in order to keep all of the pieces on the board. Black will be cramped for some period of time, but since all of the pieces remain on the board there is plenty of play left in the position.
If you just want open lines, then consider playing an early d5. For example:
1.e4 e5 2.Nf3 Nc6 3.Bc4 Nf6 4.d3 Bc5 5.c3 O-O 6.O-O d5
The downside is that lots of pieces can get traded, but black has easy development plans with Bg4, Nb6, and f7-f5 once the white bishop moves off of the diagonal.
With all that said, although black can have a hard time creating immediate imbalances, white has a very hard time getting an advantage with the Italian game, so generally keeping the pieces on the board and accepting equality will let the better player outplay his or her opponent and win.