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.
It was like a real tsunami because the whole room was upside-down [shot to look right-side up] so all that stood between us and a wave of blood was gravity. [HuffPost Canada]
Duran’s car was returned right-side up. [Examiner]
At times the movie feels like “Sideways” 30 years later and right-side up. [Boston Globe]
On the day Texas A&M’s 2014 season footballs were produced at Big Game USA’s plant, employees invited Aggies friends to sign the inside of the leather balls before they were turned right-side out. [Dallas Morning News]
The victim and his passenger were driving westbound on Magee approaching La Cholla when the black coupe pulled up on their right side. [Arizona Daily News]