It’s a bit unclear what you’re asking. But first, let’s make sure we’re understanding the idiom two sides of the same coin in the same way.
The OED’s entry for side n. offers:
c. With reference to a coin.
(a) two sides of the same coin and variants: two things, processes, etc., which are closely related or interdependent even
though they seem different; two different aspects of the same
situation or phenomenon.
Are you asking what the word is for a pair of words relative to the coin metaphor? They would indeed be — as you suggested — complements:
5. a. Something which, when added, completes or makes up a whole; each of two parts which mutually complete each other, or supply each
We need both of the words or we don’t have two sides for our “coin.”
Situated metaphorically on either side of our coin, the words are now also opposites:
1. a. Situated on the other or further side, or on either side, of an intervening line, space, or thing; contrary in position; facing.
Frequently with to and (now less commonly) from; formerly also
But maybe you are asking for a word that describes the semantic relationship between a particular pair of words?
Well, one might argue that they could all be cast as forms of opposites (if they couldn’t, we probably wouldn’t deploy them in a coin metaphor).
We have gradable antonyms — like laughter and tears — that sit on either end of a scale. They’re two sides of a human emotion coin.
We have binary antonyms — like winning and losing — where you have one or the other. They’re two sides of a game coin.
We have relational antonyms — like teacher and student — where you can’t be called a teacher if you don’t have a student and the other way around. They’re two sides of an education coin.
Ball and strike can be considered binary antonyms: pitches not hit must be one or the other. They’re two sides of the “not hit” coin. Same for fair and foul for pitches hit.
Reading is an act of intaking; writing is an opposite act — that of outputting. They’re two sides of a literacy coin.
And so on.
Definition sources: Oxford English Dictionary (login required)