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Magnetoacoustic waves in solar magnetic flux tubes may be affected by the presence of background rotational flows. Here, we investigate the behaviour of $m=0$ and $m=\pm 1$ modes of a magnetic flux tube in the presence of linear background rotational flows embedded in a photospheric environment. We show that the inclusion of a background rotational flow is found to have little effect on the obtained eigensolutions for the axisymmetric $m=0$ sausage mode. However, solutions for the kink mode are dependent on the location of the flow resonance modified by the slow frequency. A background rotational flow causes the modified flow resonances to possess faster phase speeds in the thin-tube (TT) limit for the case $m=1$. This results in solutions for the slow body and slow surface kink modes to follow this trajectory, changing their dispersive behaviour. For a photospheric flux tube in the TT limit, we show that it becomes difficult to distinguish between the slow surface and fast surface kink ($m=1$) modes upon comparison of their eigenfunctions. 2D velocity field plots demonstrate how these waves, in the presence of background rotational flows, may appear in observational data. For slow body kink modes, a swirling pattern can be seen in the total pressure perturbation. Furthermore, the tube boundary undergoes a helical motion from the breaking of azimuthal symmetry, where the $m=1$ and $m=-1$ modes become out of phase, suggesting the resulting kink wave is circularly polarised. These results may have implications for seismology of magnetohydrodynamic waves in solar magnetic vortices.