What is the largest possible area, in square centimeters, of a right triangle with one side of length 12 cm and another side of length 20 cm?
Either both given sides are legs, or the 20 cm side is the hypotenuse.  If both sides are legs, then the area of the triangle is $(12)(20)/2 = 120$ square centimeters.

If the 20 cm side is the hypotenuse, then the ratio of the given leg length to the hypotenuse is $12/20=3/5$, so the triangle is a 3-4-5 triangle and the other leg has length 16 cm.  The triangle then has area $(12)(16)/2 = 96$ square centimeters.

The largest possible area then is $\boxed{120}$ square centimeters.