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 $$u(-x_1,x_2),v(-x_1,x_2),u(x_1,-x_2),v(x_1,-x_2)$$?