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.
If some experimentator will destroy a half of his brain in such a way to not harm anything else - does he have a chance to keep being consciouness? That experiment kind of answers the question.
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.