The text behind the myth is often based on a misunderstood theory that "toilets flush in one direction in the northern hemisphere and in the opposite direction in the southern hemisphere due to the 'Coriolis Effect' ".
This myth has been perpetuated by popular television shows like the Simpsons and X-Files perhaps explaining the confusion in the public arena.
Now although the Coriolis Effect is real and measurable (due to the Coriolis Force), it is proportional to the mass of the object and in reality is so minute at a small scale compared to other factors that it is only really applicable to be described for object movements over great distances and broad segments of time, for example the movement of large weather formations in the atmosphere or water currents in the sea. Definitely not for a tiny enclosed system such as a sink drain or toilet bowl.
Generally toilets flush in the one direction constantly because of factors such as the shape of the bowl, the temperature and the angle that the water in that particular toilet enters the basin. If the stream of flushing water was encouraged to mostly enter from the opposite direction the water would then exit and spin in that opposite direction.
Because the influence of Coriolis Force is so weak the only way you are going to be able notice the Coriolis Effect in action at small scale is under very carefully controlled and averaged experimental lab conditions where all other factors of forces at play are isolated and removed from the equation.
An experiment such as this was indeed carried out and later published in the scientific journal 'Nature' without boring you with the raw data the scale of the experiment involved a large tank containing over 1000 L of water that had been allowed to settle for a full 24 hours in a temperature stabilised room before gently allowing the water to exit slowly. This experiment was duplicated and repeated in both hemispheres to confirm the action of the draining water. and so small were the forces involved that it took up to a full 15 minutes before any noticeable direction of vortex was achieved!