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On the equator, the water won't swirl, unless it is given a preferred starting direction, as in a toilet. Think of the earth as being stationary, but covered with railroad trains traveling along the parallels. The train at the equator is moving about 1000 miles per hour, while the trains at the poles are simply rotating one revolution per day. The trains moving in between vary from near zero speed to 1000 mph. If you jump from train to train from the North Pole going south, you must accelerate sideways to the right at each jump because the trains are moving ever faster. This is the Coriolis force which causes the swirl.
From the South Pole going north the acceleration is to the left. At the equator, the forces are small, since the relative speeds are similar, and there is just as much water flowing south as there is flowing north.