In this work, three different approaches are used for evaluating some Lagrangian properties of homogeneous turbulence containing anisotropy due to the application of a stable stratification and a solid-body rotation. The two external frequencies are the magnitude of the system vorticity $2\Omega$, chosen vertical here, and the Brunt–Väisälä frequency $N$, which gives the strength of the vertical stratification. Analytical results are derived using linear theory for the Eulerian velocity correlations (single-point, two-time) in the vertical and the horizontal directions, and Lagrangian ones are assumed to be equivalent, in agreement with an additional Corrsin assumption used by Kaneda (2000). They are compared with results from the kinematic simulation model (KS) by Nicolleau & Vassilicos (2000), which also incorporates the wave–vortex dynamics inherited from linear theory, and directly yields Lagrangian correlations as well as Eulerian ones. Finally, results from direct numerical simulations (DNS) are obtained and compared for the rotation-dominant case $B\,{=}\,2\Omega/N\,{=}\,10$, the stratification-dominant case $B\,{=}\,1/10$, the non-dispersive case $B\,{=}\,1$, and pure stratification $B\,{=}\,0$ and pure rotation $N\,{=}\,0$. The last situation is shown to be singular with respect to the mixed stratified/rotating ones. We address the question of the validity of Corrsin's simplified hypothesis, which states the equivalence between Eulerian and Lagrangian correlations. Vertical correlations are found to follow this postulate, but not the horizontal ones. Consequences for the vertical and horizontal one-particle dispersion are examined. In the analytical model, the squared excursion lengths are calculated by time integrating the Lagrangian (equal to the Eulerian) two-time correlations, according to Taylor's procedure. These quantities are directly computed from fluctuating trajectories by both KS and DNS. In the case of pure rotation, the analytical procedure allows us to relate Brownian $t$-asymptotic laws of dispersion in both the horizontal and vertical directions to the angular phase-mixing properties of the inertial waves. If stratification is present, the inertia–gravity wave dynamics, which affects the vertical motion, yields a suppressed vertical diffusivity, but not a suppressed horizontal diffusivity, since part of the horizontal velocity field escapes wavy motion.