They mean the same thing. They refer to an object which easily be placed in two different positions. I find it easiest to describe if I talk about a person. A person standing on his feet is "the right way up" or "right side up." A person standing on his head is "upside down." The cake in fact sounds like a "pineapple upside down cake" which is baked with the pineapple slices on the bottom of the baking dish, then turned over and served with the pineapple slices on top. It is "upside down" while it is being baked, then it is turned "right side up." The sign sounds like a sign with an arrow on it, where you might possibly have it upside down with the arrow pointing the wrong way and it wouldn't be obvious. "The right way up" and "right side up" mean the same thing. In all of these examples, "right" has the meaning of "correct." The fact that the same word is used to mean "correct" and "the opposite direction from left" is confusing in English and the source of occasional puns and jokes.

1. Turn the cake the right way up on to a wire rack. This means to make the top of the cake face upwards and the bottom side face down. The top of the cake should be the side that you can see. 2. Make sure that sign's the right way up. Right side up means to have the top or correct side facing up.This means to make sure the sign is positioned correctly. For example if the sign was turned "up side down", the bottom of the sign would be turned upward instead of the top of the sign positioned upward.