Word meaning one "half" of a dyad

Solution 1:

Monad works for dyads composed of two identical monads. In such cases, a monad is precisely one-half of such a dyad.

From Lexico:

dyad: something that consists of two elements or parts

monad: a single unit

Example:

Pollen grains may be released as monads, dyads, tetrads or polyads.’

Twin also works for dyads composed of identical parts.

If the two elements or parts of the dyad are different, you might want to consider the dyad yin and yang, yin being one part of the dyad and yang the other. Wikipedia provides the following for yin and yang:

In Ancient Chinese philosophy, yin and yang (lit. "dark-bright", "negative-positive") is a concept of dualism, describing how seemingly opposite or contrary forces may actually be complementary, interconnected, and interdependent in the natural world, and how they may give rise to each other as they interrelate to one another.

From Lexico:

dualism: The division of something conceptually into two opposed or contrasted aspects, or the state of being so divided.

Yin and yang works for many dyads (almost all?), e.g., male/female, night/day, positive/negative, winter/summer, mother/child, parter1/partner 2, etc., and you could use it in a generalized sense. One of its positive features based on your question: one can't think of yin without thinking of yang and vice versa. They embody an unseverable relationship.

Solution 2:

From dyad:

[Merriam-Webster]
1 : PAIR
specifically, sociology : two individuals (such as husband and wife) maintaining a sociologically significant relationship

As such, a dyad is a partnership, and each of of the people in such a group would be referred to as a partner:

[Merriam-Webster]
1 b : a person with whom one shares an intimate relationship : one member of a couple
      // Evan and his partner are going on a Caribbean cruise.

So, you could have a sentence like the following:

Jane is part of a dyad. She and her partner are frequently seen together.

If the sense of dyad is not as the above (the question doesn't give specific context, so I assumed the most common meaning), then use one of the following, both from Merriam-Webster.

complement:

1 c : one of two mutually completing parts : COUNTERPART
        // She is a kind of complement to me, and we get on famously.
        — Flannery O'Connor

counterpart:

2 a : a thing that fits another perfectly
2 b : something that completes : COMPLEMENT
       // the lead actress and her male counterpart