The forcing of interstellar turbulence, driven mainly by supernova explosions, is irrotational in nature, but the development of significant amounts of vorticity and helicity, accompanied by large-scale dynamo action, has been reported. Several earlier investigations examined vorticity production in simpler systems; here all the relevant processes can be considered simultaneously. We also investigate the mechanisms for the generation of net helicity and large-scale flow in the system. We use a three-dimensional, stratified, rotating and shearing local simulation domain of the size 1x1x2 kpc$^3$, forced with SN explosions occurring at the rate typical of the solar neighbourhood in the Milky Way. In addition to the nominal simulation run with realistic Milky Way parameters, we vary the rotation and shear rates, but keep the absolute value of their ratio fixed. Reversing the sign of shear vs. rotation allows us to separate the rotation- and shear-generated contributions. As in earlier studies, we find the generation of significant amounts of vorticity, with on average 65% of the kinetic energy being in the rotational modes. The vorticity production can be related to the baroclinicity of the flow, especially in the regions of hot, dilute clustered supernova bubbles. In these regions, the vortex stretching acts as a sink of vorticity. The net helicities produced by rotation and shear are of opposite signs for physically motivated rotation laws, with the solar neighbourhood parameters resulting in the near cancellation of the total net helicity. We also find the excitation of oscillatory mean flows, the strength and oscillation period of which depend on the Coriolis and shear parameters; we interpret these as signatures of the anisotropic kinetic (AKA) effect. We use the method of moments to fit for the turbulent transport coeffcients, and find $alpha_{rm AKA}$ values of the order 3-5 km/s.
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