The unsteady behaviour of a rarefied gas caused by a sudden change of the angular velocity of a sphere, placed in an otherwise quiescent gas, is investigated based on the linearized Bhatnagar–Gross–Krook model of the Boltzmann equation and the diffuse reflection boundary condition. The initial and boundary value problem is solved numerically by the method of characteristics, which is capable of tracking the discontinuity of the velocity distribution function moving in the phase space. The transient behaviour of the macroscopic quantities, such as the circumferential flow velocity and shear stress as well as the heat flow around the sphere, is clarified for a wide range of the Knudsen number. Furthermore, the long-time behaviour of the macroscopic quantities is elucidated, that is, they approach terminal values with a rate $t^{-3/2}$ for $t\gg 1$, with $t$ being a time variable. The analytical expression for the free molecular gas as well as for the slip flow is obtained. It is found that the circumferential heat flow reverses its direction as time proceeds when the Knudsen number is finite. More precisely, the direction is the same as that of the circumferential velocity of the sphere in the initial stage and opposite in the final stage, being reversed at some point of time depending on the distance from the sphere. This makes a clear contrast with the case of a free molecular gas, for which the heat flow is always in the direction of the sphere rotation in finite time and vanishes in the long-time limit.