On a German equatorial mount (Skywatcher HEQ5 or NEQ6 for example), you have a latitude adjuster which allows you to set the mount for your latitude. It goes from 0 degrees to 90 degrees and works equally well in both N and S hemispheres.
Almost. My Newtonian also has an equatorial mount with similar adjuster. On the controller there's a switch to make the drive go in the opposite direction, for use in the southern hemisphere. I've never taken it south of the equator, but I remember once flipping this switch by mistake. It took me a while to figure out why my target was constantly drifting out of the FOV!

The mount also has a built in polar scope that allows you to set the mount so it aims at either the NCP or the SCP.
Mine too. In fact I reckon that my elderly Vixen GP (still in good condition after 16 years) is more accurate than the EQ6. In Northern hemisphere use, the polaris marker on the polar scope graticule has graduations at 5-year intervals to allow for precession. Very helpful in avoiding drift!
Of course in the southern hemisphere this can't be used, because there is no equivalent to Polaris.
Back on topic, on my last trip to the southern hemisphere I took only a small refractor with non-driven mount, so the question of which hemisphere I was in, was immaterial. But I did note the fact that the Sun and other celestial objects appeared to rise, set, and transit the meridian in the opposite direction to what I'm used to.
It's amusing to see how FE'ers get around that conundrum! Of course, I know they have deeply-held beliefs about it, but...........