I am trying to understand the fundamental physics of a kind of problem. I will introduce a specific example and then list some of the broader qualms I hold. I have observed many band structures, be it spin-polarized or spin-degenerate. Let's take the case of CrI$_3$, a well-studied 2D van der Waals semiconducting magnet. The monolayer band structure is what one would expect - it has spin-polarized carriers at the band edges, i.e. the majority and minority carriers occupy different energy eigenstates. Moving onto the bilayer, however, the band structure is spin-degenerate. Many papers like this one  report PT-symmetry (Parity-Time) as the cause for the degeneracy: basically, the Hamiltonian commutes with the product of P$\cdot$T. My main questions, which is not just specific to CrI$_3$, are as follows:
Is there a rigorous way when one looks at new material, to say if PT-symmetry would lead to a spin degeneracy? Because from what I've read, and I could totally be wrong, but it looks like people do band structure calculations and then attributes such a degeneracy to PT-symmetry.
I do not hold a good theoretical physics footing, so I don't know much about PT symmetry with regard to specific systems, but I do know about the universal CPT symmetry that is preserved. Is one just a subset of the other?
Finally, I am having a hard time trying to understand how the system can break inversion symmetry and time-reversal symmetry but the product of these two symmetry operators is still a 'good' quantum number i.e. one that commutes with the Hamiltonian.
Monolayer CrI$_3$ with spin-polarized bandstructure:
Bilayer CrI$_3$, where spin carriers are degenerate:
PS: I am well aware that the answer might involve a thorough level of detail, so I'm very open to awarding bounty points for detailed answers.