More like a re-invent but a symmetrical surfboard. It would be the invader of surfboards I guess. It would either need a very dull point on both ends or some type of broad curve that still allows it to bite the wave. A normal pointed nose shape on both ends doesn't seem ideal. Symmetrical rocker, but not too much and it would need to be gradual. It seems like too much rocker would make it way too loose. Fins wouldn't work, not on the back anyway. Central fins or several channels to well, channel water through and keep it on course would be put into place. The fin would need to be symmetrical too. I have no clue where the leash would go.
I know people do reverts and stuff in waves, but this could improve on it. Also, a board with a lot of rocker would behave loosely and kinda like a skateboard in the sense that you could nearly ollie with it. I guess two models, one with a lot of rocker and another the opposite, would be made.
Why I think it won't work: Wikipedia says surfing has been around since 1767, so I'm probably not the first one to think of this. That means it may have been tried and failed or labeled as non-beneficial no matter how the board is made. Then again, skiing's been around for a while and symmetrical only caught on a few years ago...