The question is somewhat complex and directed to clearing things out.
Suppose that $n$ is the order of the cyclic group. It $n - 1$ is the number of all private keys possible
n = 0xfffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141
We also know that every private and public key has its modular inverse. To get a modular inverse of a private key, we need to subtract the private key from $n$.
$$n - privKey$$
To get a modular inverse of a public key, we'll have to multiply its $y$ coordinate by $-1$ and modulo by the $p$ - order of the finite field. $$x,y = x, -y \bmod p$$
A modular inversed public key has the same $x$ coordinate as the original public key, but a different $y$ coordinate, and the $y$ coordinate is always different in its polarity. If the original $y$ was odd, in a modular inversed key it will be even, and vice versa.
The question is, if the $y$ coordinate of a public key is even, does it mean that the corresponding private key is less than $n/2$ by its value? If the $y$ is odd, the private key is more than $n/2$?
Is there any relationship between the evenness/oddness of the $y$ (or $x$) coordinate and the value of the corresponding private key?
Is there any way to know that the private key is more or less than $n/2$ while not knowing the private key itself?