  The discrete symmetries of the Lorentz group are on the one hand a `complex'
interplay between linear and anti-linear operations on spinor fields and on the
other hand simple linear reflections of the Minkowski space. We define
operations for T, CP and CPT leading to both kinds of actions. These operations
extend the action of SL(2,C), representing the action of the proper
orthochronous Lorentz group SO^+(1,3) on the Weyl spinors, to an action of the
full group O(1,3). But it is more instructive to reverse the arguments. The
action of O(1,3) is the natural way how SL(2,C) together with its conjugation
structure acts on Minkowski space.
  Focusing on the symmetries of these (anti-)linear operations we can for
example distinguish between CP-invariant and CP-violating symmetries. This is
important if gauge symmetries are included. It turns out that, contrary to the
general belief, CP and T are not compatible with SU(n) for n > 2, especially
with colour-SU(3) or with the U(3)-Cabibbo-Kobayashi-Maskawa matrix.
