The problem of determination of the upper critical value and the form of the loss of stability during torsion of a shell of revolution of negative Gaussian curvature is considered. A method of asymptotic integration developed by T. V. Liyva is used. It is taken into account that initial stresses and radii of shell curvature are variable functions of the length of the arc of the generatrix. Cases are also investigated where the middle surface of the shell permits non-trivial deflections. As an example, torsion of a part of a torus with a negative. Gaussian curvature is considered.