A C*-algebra is symmetric if it is isomorphic to its opposite algebra, or equivalently if it has a conjugate-linear automorphism. Not all nuclear C*-algebras, not even all homogeneous C*-algebras, are symmetric, but all currently classifiable ones are. A sufficiently nonsymmetric C*-algebra (one not KK-equivalent to its opposite algebra) would even be a counterexample to the UCT. A better understanding of symmetry will be needed to extend classification beyond its current limits, especially to nonsimple C*-algebras. I will discuss some aspects of symmetry and th rigidity of complex structure in C*-algebras, with some examples. The talk will be rather speculative, with more questions than answers.