  We prove a remarkable combinatorial symmetry in the number of spanning
configurations in site percolation: for a large class of lattices, the number
of spanning configurations with an odd or even number of occupied sites differs
by $\pm 1$. In particular, this symmetry implies that the total number of
spanning configurations is always odd, independent of the size or shape of the
lattice. The class of lattices that share this symmetry includes the square
lattice and the hypercubic lattice in any dimension, with a wide variety of
boundary conditions.
