Parallel transport around closed circular orbits in the equatorial plane of the Taub–NUT spacetime is analysed to reveal the effect of the gravitomagnetic monopole parameter on circular holonomy transformations. Investigating the boost/rotation decomposition of the connection 1-form matrix evaluated along these orbits, one finds a situation that reflects the behaviour of the general orthogonally transitive stationary axisymmetric case and indeed along Killing trajectories in general.