If I have a complex valued odd/even holomorphic function, can I say anything about parity of its real/imaginary part as $R^2$ functions?

Let $f(z)=f(x_1,x_2)=f(x_1+i x_2)=u(x_1,x_2)+iv(x_1,x_2)$ be a holomorphic function, then by Cauchy-Riemann:
$frac{partial u}{partial x_1}(x_1, x_2)=frac{partial v}{partial x_2}(x_1, x_2), quad
frac{partial u}{partial x_2}(x_1, x_2)=-frac{partial v}{partial x_1}(x_1, x_2).$

If I’m not wrong it means that if $u$ is odd in $x_i$, $v$ must be even in $x_i$, right?

My question is: suppose $$f(-z)=-f(z),$$
then it must be true that $u(-x_1,-x_2)=-u(x_1,x_2),v(-x_1,-x_2)=-v(x_1,x_2)$. But,

can I say anything about $u(-x_1,x_2),v(-x_1,x_2),u(x_1,-x_2),v(x_1,-x_2)$?