commutator - A question about causality and Quantum Field Theory from improper Lorentz transformation

Related post Causality and Quantum Field Theory

In Peskin and Schroeder's QFT p28, the authors tried to show causality is preserved in scalar field theory.

Consider commutator

$$[ \phi(x), \phi(y) ] = D(x-y) - D(y-x) \tag{2.53}$$ where $D(x-y)$ is the two-point correlation function, $$D(x-y):= \langle 0 | \phi(x) \phi(y) | 0 \rangle = \int \frac{d^3 p}{ (2\pi)^3} \frac{1}{ 2E_{\mathbf{p}}} e^{-ip(x-y)}\tag{2.50}$$

P&S argued that each term in the right-hand-side of (2.53) is Lorentz invariant, since

$$\int \frac{d^3p }{ (2\pi)^3} \frac{1}{2E_{\mathbf{p}}} = \int \frac{ d^4 p }{ (2\pi)^4} (2\pi) \delta(p^2-m^2)|_{p^0>0} \tag{2.40}$$ is Lorentz invariant.

Since there exists a continuous Lorentz transformation in the spacelike interval $(x-y)^2<0$ such that $(x-y) \rightarrow - (x-y)$ and $D(y-x)=D(x-y)$, (2.53) equals zero in the spacelike interval. In timelike interval, since such continuous Lorentz transformation does not exist, (2.53) is non-zero in general.

My question is, consider a non-continuous Lorentz transmation in the timelike interval, $PT$, namely time reversal times parity transformation. I can also let $(x-y) \rightarrow - (x-y)$. Why (2.53) in the timelike interval is non-zero?

I guess $PT$ will let (2.40) go to $p^0<0$ branch. But I am not sure if it breaks the Lorentz invariant of (2.40) and (2.50).