I'm reading through Digital Signal Processing, Proakis and Manolakis, third edition. I've reached section 3.2: Properties of $\mathcal Z$-transform.
One property is the convolution:
$$x(n) = x_1(n)\star x_2(n) \longleftrightarrow X(z) = X_1(z)X_2(z)$$
Other property is correlation:
$$r_{x_1,\ x_2}(l) = \sum_{n=-\infty}^{\infty}x_1(n)x_2(n-l) \longleftrightarrow R_{x_1,\ x_2}(z)=X_1(z)X_2\left(z^{-1}\right)$$
I'm confused by the time reversal (folding) which appears in the correlation property at $X_2\left(z^{-1}\right)$. I know that the folding step occurs on convolution but not on cross/auto/correlation. Why is this and what am I missing?
To clarify the "time reversal" expression I'm referring to another property of $\mathcal Z$-transform taken from the same source:
If $$x(n) \longleftrightarrow X(z)$$ Then $$x(-n) \longleftrightarrow X(z^{-1})$$
For the two answers: Matt's one clarifies the matter using a mathematical perspective while Peter's one shows in an intuitive way how the folding occures in polynomial multiplication. So thanks, I wish SE had a way to mark two answers as accepted.