Although very unlikely, the starting position could be a forced win for black. This would mean it is a position of mutual zugzwang, in the strong sense that whoever moves first loses. Wikipedia (Link) refers to one such position as trébuchet. Such positions are rare, and it would be a big surprise if the starting position was one of them.
The starting position has another property: it is symmetric. White and black have exactly the same possible moves (when written in descriptive notation).
My question: Are there any symmetric positions that are known to be a mutual zugzwang?