  We classify birational involutions of the real projective plane up to
conjugation. In contrast with an analogous classification over the complex
numbers (due to E. Bertini, G. Castelnuovo, F. Enriques, L. Bayle and A.
Beauville), which includes 4 different classes of involutions, we discover 12
different classes over the reals, and provide many examples when the fixed
curve of an involution does not determine its conjugacy class in the real plane
Cremona group.
