Photonic topological insulator plates violate Lorentz reciprocity, which leads to a directionality of surfaceguided modes. This in-plane directionality can be imprinted via an applied magnetic field. On the basis of macroscopic quantum electrodynamics in nonreciprocal media, we show that two photonic topological insulator surfaces are subject to a tunable, magnetic-field-dependent Casimir torque. Due to the directionality, this torque exhibits a unique 2π periodicity, in contradistinction to the Casimir torques encountered for reciprocal uniaxial birefringent media or corrugated surfaces which are π periodic. Remarkably, the torque direction and strength can be externally driven in situ by simply applying a magnetic field on the system, and we show that this can be exploited to induce a control of the rotation of small objects. Our predictions are relevant for nanooptomechanical experiments and devices.