Are there any symmetric zugzwang positions? 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 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 in that it is symmetric. White and Black have exactly the same possible moves (when written in descriptive notation). Are there any symmetric positions that are known to be a mutual zugzwang?
You can easily construct many such positions. For instance:

4k3/8/3P1P/8/8/3p1p2/8/4K3 w - - 0 1


First to move loses - either by moving a king and allowing the opponent to promote or by losing one (and subsequently next) pawn.