Parity and time reversal symmetries in QFT and the Standard Model The parity transformation $\mathcal{P}$ and the time-reversal transformations $\mathcal{T}$ are defined as follows :
\begin{equation}
\mathcal{P}=
\begin{bmatrix}
1 & & & \\
 & -1 & & \\
 & & -1 & \\
 & & & -1
\end{bmatrix},\qquad
\mathcal{T}=
\begin{bmatrix}
-1 & & & \\
 & 1 & & \\
 & & 1 & \\
 & & & 1
\end{bmatrix}.
\end{equation}
Those transformations are clearly element of the Poincaré group, and in particular of the Lorentz group but they are no restricted Lorentz transformation (or proper orthochronous Lorentz transformation). In QFT, Lorentz symmetries $\Lambda$ (and in general any symmetries) are implemented through the use of unitary/antiunitary operators $U(\Lambda)$. Those act as
\begin{equation}
U(\Lambda)A_\mu(x)U(\Lambda) = \Lambda_\mu^{~~\nu}A_\nu(\Lambda x)
\end{equation}
on a quantum vector field $A$ for example. We usually use the notations $P\equiv U(\mathcal{P})$ and $T\equiv U(\mathcal{T})$. Since QFTs are relativistic theories, they must be invariant under the Poincaré group and so in particular under the Lorentz group (proper and orthochronous or not).
Therefore, I don't understand why we can consider Lagrangian densities $\mathcal{L}$ that are not invariant under $P$ and $T$ (more precisely, Lagrangians that leads to actions that are not invariant), why is that? For example, it's a well known fact that electroweak interactions
\begin{equation}
\mathcal{L}_{EW}=\bar{\psi}_LW_\mu\gamma^\mu\psi_L = \frac{1}{2}W_\mu(\bar{\psi}\gamma^\mu\psi-\bar{\psi}\gamma^\mu\gamma^5\psi)
\end{equation}
are not invariant under $P$, where $\gamma^\mu$ are Dirac's matrices, $\psi$ is a fermionic field and $W$ is the $SU(2)_L$ gauge boson.
 A: Just as the proper orthochronous Lorentz transformations form a subgroup, the same is true for the Poincare transformations.
You can't reverse the time or reflect the space by any combination of the proper orthochronous Lorentz transformation and shift. So $\mathcal{P}$ and $\mathcal{T}$ don't belong to the connected component of the Poincare group that is generated by exponentiation of its Lie algebra. In fact the full Poincare group of your preference has similar structure to the Lorentz group - the semidirect product of a proper orthochronous Poincare group and a discrete group $\{1,\mathcal{P},\mathcal{T},\mathcal{PT}\}$.
There's no inconsistency in violating the discrete symmetries and everything about special relativity kinematics and dynamics remains to be true. Your statement that "relativistic theories ... must be invariant under the Poincaré group and so in particular under the Lorentz group (proper and orthochronous or not)" is simply too restrictive.
Your mention that the Lorentz symmetries are represented by unitary seems irrelevant but note that $U(\mathcal{T})$ is not unitary, rather it is antiunitary (i.e. it is an antilinear operator that conserves the inner products)
