We study the possibility of toroidal twist-like bifurcations for an isotropic Levinson–Burgess compressible elastic tube subject to a pure circular shear. We intend to model forms of instability for solids that are analogous to the classical Taylor–Couette patterns observed in the flow of viscous fluids. We first establish that the axisymmetric circular shear deformation is a fundamental solution of the equilibrium problem, and then investigate the possibility that this primary deformation may bifurcate into an axially periodic toroidal twist-like mode by analyzing the related incremental boundary-value problem. The analysis of the bifurcation problem and the evaluation of the critical load are carried out by following a novel effective procedure, based on the Magnus expansion.