The noun phrase right-side up describes a direction in which the correct side of an object is facing up. Right-side is a hyphenated compound noun, while up completes the noun phrase. In this phrase the word right has the definition of being correct, not the opposite of left.
The same is true for the phrase right-side out, and the nouns right-hand and left-hand. Though when you are speaking about the right side of an object, meaning the opposite of the left side, there is no hyphen. The distinction arises between the direction being fixed (right-side) or variable depending on your point of view (e.g., the right side for me could be the left side for you). This variable definition extends to the distinction between right and wrong (e.g., the right side of the argument).
There is some transition happening with hyphens in general, with most everyone leaning to the elimination of them. It is easy to see this phrase merging to one word in the future similar to upside, but as for now use the hyphen.