An axisymmetric contact problem concerning the torsion of a circular shaft of an orthotropic-nonhomogeneous half-space is considered. By means of the technique of integral transformations of Laplace and Hankel, with the subsequent application of the orthogonal polynomial method, an approximate solution in the transformant space is constructed. Also was performed reverse transformation. Calculated formulas for the angle of rotation of the shaft and the tangential stress acting on the contact area are obtained. Numerical calculations for certain types of heterogeneity have been performed. Comparison of the obtained results with the previously known results is made.