A birational map of the (affine or projective) plane is an isomorphism between complements of algebraic curves. Over the complex numbers, the classification of birational involutions of the plane is particularly nice: the conjugacy classes of involutions fixing an irrational curve are completely determined by the isomorphism class of the fixed curve. This is false over the real numbers. I will present the classification and infinitely many non-conjugate involutions fixing isomorphic fixed curves.