How does Power Spectrum remain symmetric in Z domain?

Can you tell me how the $$P_x(z)=P_x^*(1/z^*)$$ is mathematically correct. I can understand the $$P_x(e^{jw})=P_x^*(e^{jw})$$ as $$P_x$$ is real value. But why take the Z domain representation in this way ($$P_x(z)=P_x^*(1/z^*)$$). Why can't I take it like $$P_x(z)=P_x^*(z)$$ like the Fourier transform representation?

The condition $$P_x(e^{j\omega})=P^*_x(e^{j\omega})$$ corresponds to the time domain symmetry

$$r_x(n)=r^*_x(-n)\tag{1}$$

where $$r_x(n)$$ is the autocorrelation of $$x(n)$$, which is the inverse (discrete time) Fourier transform of $$P_x(e^{j\omega})$$.

Now you just need to figure out the $$\mathcal{Z}$$-transform of $$r_x^*(-n)$$ in terms of $$P_x(z)$$ in order to see the symmetry relationship in the $$\mathcal{Z}$$-transform domain:

\begin{align}\mathcal{Z}\big\{r_x^*(-n)\big\}&=\sum_nr_x^*(-n)z^{-n}\\&=\sum_nr_x^*(n)z^n\\&=\left[\sum_nr_x(n)(z^*)^n\right]^*\\&=\left[\sum_nr_x(n)\left(\frac{1}{z^*}\right)^{-n}\right]^*\\&=P_x^*\left(\frac{1}{z^*}\right)\tag{2}\end{align}

Consequently, the symmetry condition $$(1)$$ implies

$$P_x(z)=P_x^*\left(\frac{1}{z^*}\right)\tag{3}$$

For real-valued sequences, $$(3)$$ can also be written as

$$P_x(z)=P_x\left(\frac{1}{z}\right)\tag{4}$$

Simple way of seeing this:

From the definition of the Fourier Transform, when you transform a signal into a linear combination of sines (or cosines), you get the complex conjugates of all the possible values within the spectrum of the input signal. I am not an expert on mathematics but, when you transform the exponential term of your Fourier transform into cosines and sines, you have two sides, a positive side of your spectrum which are all the real values and the negative side of your spectrum which are all the complex values.

As you see here, they have the same value in magnitude but different signs, that is why you have symmetry with respect to zero (positive and negative).

Please, correct me if I am wrong! But this is a simple way to explain that symmetry. I am sure someone else could give a bit more detailed explanation