Percolation is perhaps the simplest non-trivial model in statistical mechanics, but has remained under active study for more than forty years. In 2-D it exhibits a second-order phase transition, at which a number of interesting and little-understood symmetries manifest themselves. We discuss three of these: (a) the horizontal crossing probability, which reveals a triangular symmetry, (b) an exact factorization of certain correlation functions, and (c) a generalization of this factorization that shows a mysterious independence of one coordinate. We demonstrate (c) via the explicit calculation of a certain six-point correlation function. Both (b) and (c) generalize to a variety of other two-dimensional critical points. The main tool employed is conformal field theory.