The influence of background rotation on all nontrivial third-order vorticity correlations is studied for an unbounded incompressible homogeneous turbulent flow, using pseudo-spectral direct numerical simulation. The behaviour of third-order vorticity correlations is found to be consistent with exact theoretical predictions presented herein for axisymmetric turbulence without mirror symmetry. Particular attention is given to the vertical vorticity skewness S_omega3. Its dependence on the viscosity, the initial value of the velocity gradient skewness, and the background rotation rate has been thoroughly investigated. The initial growth rate of S3 provides evidence for a power-law behaviour proportional to t^(0.75±0.1) for all considered cases, in agreement with recent experimental results by Morize et al. [Phys. Fluids 17 (2005) 095105]. It is also found that higher background rotation rates - implying more linearity - result in lower final values of while lower viscosities and lower initial (absolute) values of the velocity derivative skewness both yield higher final values of S_omega3.