It is not selected; it is computed.
The elliptic curve point multiplication algorithm outputs a point, which has an X and Y coordinate.
While it is true that negating the private key will negate the public key’s Y coordinate, there is no simple rule for determining what the parity of Y will be without actually performing the elliptic curve multiplication operation. In fact, there cannot be such a rule; it would constitute a break in the discrete logarithm security of the curve.
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