The key is in finding a way to break the symmetry. For example, you could play a move where if they copy you they're hanging a piece. Or make a checking move - then they're forced to do something different. And if you make a capture with the check, then they might not be able to regain the symmetry afterwards.
Black mirrors my moves
The position is not always equal - it's really easy to set traps, i.e.
Quite vague brah
Sorry wafflemaster, but 1random is best brah. But, I will continue to copycat if I am black and this is done, brah. Becuase it is all about principle.

Many openings can start off symmetrical, but copying for too long is a blunder.
For example, Spanish Four Knights:
1.e4 e5 2.Nf3 Nf6 3.Nc3 Nc6 4.Bb5 Bb4 5.O-O O-O 6.d3 d6 7.Bg5, here 7...Bg4 is a blunder and White should play 7...Bxc3. I don't recall totally as I don't play this opening, but I want to say that 7...Bg4 is answered by 8.Nd5, but don't quote me on it, I'm working off memory.
Has anyone ever played someone where they mirror your moves? Any traps that is possible to set? Maybe not, because the position is always equal.