QFT: Vacuum invariant, but vacuum correlations aren't

Consider a free scalar field theory. My struggle is that vacuum correlation functions of fields are only Lorentz invariant under a subgroup of Lorentz transformations, despite the invariance of the vacuum under the complete group of Lorentz transformations! I expect that I am making suspect assumptions somewhere.

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I expect the vacuum to be invariant under more than just proper, orthochronous Lorentz transformations: I expect the vacuum to be invariant under time reversal invariance and spatial inversion: $T|0rangle = |0rangle$ and $P|0rangle = |0rangle$, where these operators act on the field operators as $T^{-1} psi(x) T = psi(Lambda_Tx)$ and $P^{-1} psi(x) P = psi(Lambda_Px)$ where $Lambda_T$ and $Lambda_P$ are the usual 4×4 matrices for time reversal and inversion.

However, vacuum invariance implies invariance of correlation functions: consider
begin{align*}
D(x,y) &= langle 0| psi(x) psi(y) |0rangle &= langle 0|P^{-1}P psi(x) P^{-1}P psi(y) P^{-1}P|0rangle &= langle 0|psi(Lambda_Px)psi(Lambda_Py)|0rangle &= D(Lambda_Px, Lambda_Py)
end{align*}

This holds similarly for $T$, $D(x,y) = D(Lambda_Tx,Lambda_Ty)$.

However, (see below) I don’t think $D(x,y) = D(Lambda_Tx,Lambda_Ty)$ is true!

The fact that $D(x,y) = langle 0| psi(x) psi(y) |0rangle$ is only invariant ($D(Lambda x, Lambda y) = D(x,y)$) under proper, orthochronous Lorentz transformations and not generic Lorentz transformations comes up in discussing causality. Invariance under proper, orthochronous transformations means the commutator $[psi(x),psi(y)]$ will vanish for spacelike $x-y$, which it does. Invariance under all transformations would mean the commutator would vanish for timelike $x-y$, but it doesn’t! See also A question about causality and Quantum Field Theory from improper Lorentz transformation for background.

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What am I getting wrong?

My guesses for what’s wrong above:

1. The vacuum is not invariant under time reversal and spatial inversion. Seems unlikely to me.
2. The fields transform differently under the operator implementations of $T$ and $P$. Seems unlikely to me.
3. My insertions of $I = P^{-1} P$ and $I = T^{-1} T$ are mistaken, perhaps in the latter case by the anti-unitarity of the operator implementation of $T$. Unsure.