The fact known from ab initio calculations, that in molecular systems with real Hamiltonian, surrounding a two-state conical intersection (ci) with a small circle in the configuration space of nuclear coordinates the angle of the adiabatic-to-diabatic transformation (ADT) matrix forms a phase of +/-pi, is rigorously proved for the most general case and a condition is given to decide whether +pi or -pi values should be assigned to the particular seam of ci. We show how can be explained on a straightforward way the disappearance i.e. the 'pair annihilation' of conical intersections with the aid of sign attached to the ci. (C) 2004 Elsevier B.V. All rights :reserved.