Let me try to clear things out. You are mixing the symmetries of spacetime with the symmetries of the interactions of the Standard Model.
The first, are symmetries of spacetime to which according to special relativity (in relation to the Standard Model) is the Poincaré group of symmetries. So spacetime itself is translationally invariant, isotropic, parity symmetric, time symmetric (this actually just a case of translation in the time coordinate).
A different set of symmetries are those associated to the fields and the interactions themselves. Namely, to the models we build on spacetime. For this we use fields, they correspond to forces and to matter, but they have different internal symmetries, technically they are associated to irreducible representations depending on the types of charges they carry.
Now here is what is more specific. $C, P$ and $T$ are symmetries that speak about the fields and how they transform within a model. Sadly $P$ and $T$ already appear in the spacetime symmetries therefore the confusion, but what people mean is their action on the fields and the interactions of the theory.
So they are two different things under study (with a similar name). The symmetries of a theory do not have to be the same as the symmetries of the spacetime it is built on.
P.D. As soon as you assume rotations and translations your spacetime is isotropic and homogeneous so will also be symmetric under reflection.