Some special problems for axisymmetric solids made of linearly elastic orthotropic micropolar material with central symmetry are dealt with. The first one is a hollow circular cylinder of unlimited length, subjected to internal and external uniform pressure. The second one is a hollow or solid circular cylinder of finite length, subjected to a relative rotation of the bases about its axis. In both cases, one of the axes of elastic symmetry is parallel to the cylinder axis; the other two are arbitrarily oriented in the plane of any cross-section of the solid. The elastic properties are invariant along the cylinder axis. It is shown that the two problems are governed by formally similar sets of ordinary differential equations in the kinematic fields (in-plane displacements and microrotations). In the general case, numerical solutions are derived. The solution for the cylinder subjected to radial pressure does not significantly differ from that obtained in classical elasticity, at least in terms of radial and hoop force stresses. In the case of a cylinder subjected to torsion the difference between the micropolar and the classical solutions is more pronounced. The torque induces twisting couple stresses about the cylinder axis of variable sign. Finally, size effects in terms of torsional inertia are pointed out.