Thought experiments are very common in Special and General Relativity (SR,GR). Usually a thought experiment is structured as follows:

1. We present a setup in a frame of reference
2. We apply principles of SR or GR to derive what must happen
3. We then shift perspective to another frame of reference
4. We state that both observers should agree on what's happening
5. We at last draw the conclusions

This structure lets us derive directly from the postulates effects like Time Dilation, Light Bending in a Gravitational Field and so on.

My problem regards the point 4.: In every thought experiment we state that observers should agree on what is happening in some way. Problem is that this is not always true! If I state that observers in different frames should agree on simultaneity, for example, I am obviously in the wrong. But if I state that both observers should see a mug breaking I am probably correct.

My question is: The fact that both observers, in different frames, should agree on events is a principle? If so what is the precise wording of this principle? This principle has a name?