  The purpose of this short paper is to investigate relations between various
real K-theories. In particular, we show how a real projective bundle theorem
implies an unexpected relation between Atiyah's KR-theory and the usual
equivariant K-theory of real vector bundles. This relation has been used
recently in a new computation of the Witt group of real curves by Schlichting,
Weibel and the author. We also interpret Atiyah's theory as a special case of
twisted K-theory.
