Your best bet is to talk to your actual lesson teacher and discuss with them what type of sound you want and what would best support you getting there.
If you're looking for a dark, orchestral sound, you usually need a somewhat wider mouthpiece which tends to come with a deeper cup and open backbore.
If you're a jazz player or frequently playing baroque music so you NEED the altissimo register, also understand that those "cheater" mouthpieces don't make the notes EASIER to play, they just provide more support for your ability to play them. Your own musical skill and muscle development is needed to buzz those high notes. The mouthpiece won't improve your range.
I've never played an asymmetric mouthpiece.
I've played a Denis Wick brand Maurice Murphy 3C now for a decade - and only because I don't play enough anymore to need a 1C. That's simply the most comfortable mouthpiece I've ever owned. Great sound, great support, great "cushion" on the lips, and I can play anything I need to play.