The paper presents results for the resolved numerical simulation of a turbulent flow past a homogeneous sphere and a spherical shell of equal mass and radius (and, therefore, with a larger moment of inertia) free to rotate around a fixed center. This situation approximates the behavior of a particle whose relative motion with respect to the fluid is driven by external forces, such as a density difference in a gravitational field. Holding the center fixed makes it possible to have precise information on the turbulent flow incident on the particle by repeating the same simulations without the particle. Two particle Reynolds numbers based on the mean velocity, Rep=80 and 150, are investigated; the incident turbulence has Reλ=36 and 31, respectively. The particle diameter is an order of magnitude larger than the Kolmogorov length scale and close to the integral length scale. The turbulent eddies that interact most strongly with the particle are characterized. Their size is found to increase with Rep due to the interplay of the convection timescale, the particle timescale, and the eddy timescale, but it remains of the order of the particle diameter. The sign of the hydrodynamic torque is likely to persist much less than the convection time, although longer durations are also found, revealing the effect of occasional interactions with larger eddies. The autocorrelation of the torque changes sign at shorter and shorter fractions of the convection time as the Reynolds number increases. Significant cross-stream forces are found. An analysis of their magnitude shows that they are mostly due to induced vortex shedding combined with a weaker Magnus-like mechanism.