Geophysical flows are often turbulent and subject to rotation. This rotation modifies the structure of turbulence and is thereby expected to sensibly affect its Lagrangian properties. Here, we investigate the relative dispersion and geometry of pairs, triads and tetrads in homogeneous rotating turbulence, by using direct numerical simulations at different rotation rates. Pair dispersion is shown to be faster in the vertical direction (along the rotation axis) than in the horizontal one. At long times, in Taylor’s regime, this is due to the slower decorrelation of the vertical velocity component as compared to the horizontal one. At short times, in the ballistic regime, this result can be interpreted by considering pairs of different orientations at the release time, and is a signature of the anisotropy of Eulerian second-order functions. Rotation also enhances the distortion of triads and tetrads also present in homogeneous and isotropic turbulence. In particular, at long times, the flattening of tetrads increases with the rotation rate. The maximal dimension of triads and tetrads is shown to be preferentially aligned with the rotation axis, in agreement with our observations for pairs.