Yeah yeah, of course cyclone is not spinning the same way in NH and SH. And if you dig in the etymology of the word cyclonic and anticyclonic, it is quite two strange words. Cyclone comes from Greek and does not implies a priori a specific direction for the spin. Etymologically speaking, we could make a case that anticyclonic is cyclonic... It is worth remembering also that we are speaking about vorticity along z-axis. Vorticity along x and y axis are negligible in general ( excepted in tornadoes XD ). And, mathematically speaking, positive vorticity is counter-clockwise, and negative vorticity is clockwise. This said, when I speak about a cyclone, for me it is a positive spin in NH and negative spin in SH. Of course, there is a link between the vorticity and the local pressure anomaly. If you consider p', the field of the pressure anomaly (anomaly regarding a space mean p0, not a time mean), you can write something like this -k and v vector- :
-1/ρ0 * grad( p' ) = f0 * k ⋀ v
Or more simply, in NH positive spin around a low pressure anomaly (1025 hPa can still be cyclonic if embedded in a 1050 hPa anticyclonic rotation ...) and a negative spin around a low pressure anomaly (again, 1000 hPa can be quite anticyclonic). I'm note sure if there is a best way to describe this equation. Coriolis force and pressure gradient act together and you have something like a double entry table. If NH and convergent flow, positive vorticity. If SH and convergent flow, negative vorticity. If NH and divergent flow, negative vorticity. If SH and convergent flow, positive vorticity.