It is shown that Co-rich microwires with definitely negative magnetostriction have nearly linear axial magnetization cur-ves with a very small remanence, because they are magnet-ized circumferentially in the absence of an applied magnetic field. For these wires a complete set of material parameters, i.e. saturation magnetization, magnetostriction constant and characteristic amplitude of the residual quenching stress is obtained by means of careful measurement of the wire re-sponse under tensile stress to slowly varying applied magne-tic field. Using the same technique we demonstrate also the existence of a longitudinally magnetized core for Co-rich wires having hysteresis loops with finite remanence. The fi-nite remanence for these wires is explained assuming the magnetostriction coefficient to be radial dependent, so that it may change sign in a certain internal point along the wire ra-dius.