Clearly not, since you can have consciousness with only half a brain. Even a full hemispherectomy doesn't make you not conscious. In fact, there may be little change to the personality or cognition at all.

The symmetry between right and left brain doesn't really correspond to the types of symmetries they're talking about in the article. They're usually expressed as "The laws of physics are independent of X; you can substitute X with Y and the laws work the same." There isn't any kind of substitution here, so no place to apply the symmetry.

Given that the left and right brain aren't exactly symmetrical and will receive and process different inputs that seems like a no.

In my opinion, one is between "this needs to be remembered, I will need this later" and "I have already remembered this, immediately recover!".