Suppose I have a QFT defined by a Lagrangian in Minkowski space and one in Euclidean space related by a Wick Rotation. What sort of objects/properties in general stay the same between either theory; and when am I allowed to perform one?

I am aware of the reconstruction theorem that somehow relates Euclidean path integrals to Lorentzian ones. However, that in general only seems to talk about the correlation functions.

I am wondering this since I just learned that it's difficult to canonically quantize the Maxwell Lagrangian due to the inconsistent choices of "Norms". I was wondering if it would be possible to Wick Rotate to Euclidean Space, Quantize and then go back. However, I was informed that this would not lead to the correct set of particles for the theory (Different Irreps). So, I am trying to understand what in general stays the same between Lorentzian and Euclidean theories and what doesn't.