If $x-y$ were written as $-y+x$, then would its conjugate be $-y-x$? The conjugate of $x-y$ is $x+y$. But, if I wrote it as $-y +x$, would the conjugate be $-y-x$?
Furthermore, is the conjugate of $x + y +z$

*

*$(x+y)+z \to (x+y) - z $


*$(x+z)+y \to (x+z) - y $


*$(z+y)+z \to (z+y) - x $
Is there a way to formalize the idea of a conjugate?

The term "conjugate" is meant to be used with numbers, where in some context there is a natural other number that they pair with. Usually, it's pairs of numbers that are roots of the same quadratic polynomial with coefficients in a more restrictive field.
So when $a,b$ are real, $a+bi$ and $a-bi$ are conjugates because they are both roots of $x^2-2ax+(a^2+b^2)$, a quadratic polynomial with real coefficients. It's not really about changing a plus sign to a minus sign.
Similarly when $a,b$ are rational, $a+b\sqrt{2}$ and $a-b\sqrt{2}$ are conjugates because they are both roots of $x^2-2ax+(a^2-2b^2)$, a quadratic with rational coefficients. Again, it's not really about changing a plus sign to a minus sign.
So if you rearrange to $bi+a$, it still has the same conjugate: $a-bi$ (or if you want to write it as $-bi+a$, fine.) Because $bi+a$ and $a-bi$ are the two roots of the same polynomial.
And with abstract $x$ and $y$ without more context, there is not a good meaning for the conjugate of $x+y$.
