Why if we rotate an object by R and then translate it with T, from the object perspective, we're actually applying $-T$ and then $-R$ I've been introduced to local coordinates in contrast to the global ones which I was used to work on, where we have basically the camera at the origin.
For example, I've the following picture of a cube:

According to a slides that I've:

the object is rotated about the x-axis by $R$ and then translated by $T$
from the object's perspective, the world is first translated by $–T$ and then rotated about the (object's local) x-axis by $–R$

Not just the sign but also the order of the transformations changes. Why is that?
Note: I've some ideas, clearly, but I would like to hear a good answer with a mathematical explanation.
If you have object$\rightarrow$world space transformation matrix:
$$M=T*R$$
then inverse (i.e. world$\rightarrow$object) of this transformation matrix is:
$$M^{-1}=(T*R)^{-1}=R^{-1}*T^{-1}$$
So the order of multiplication of matrix inverses is reversed per matrix inversion rules.