  We show that every automorphism of a thick twin building interchanging the
halves of the building maps some residue to an opposite one. Furthermore we
show that no automorphism of a locally finite 2-spherical twin building of rank
at least 3 maps every residue of one fixed type to an opposite. The main
ingredient of the proof is a lemma that states that every duality of a thick
finite projective plane admits an absolute point, i.e., a point mapped onto an
incident line. Our results also hold for all finite irreducible spherical
buildings of rank at least 3, and as a consequence we deduce that every
involution of a thick irreducible finite spherical building of rank at least 3
has a fixed residue.
