Question: Torsion is usually described as a rotation of the tangent vector along the geodesic, like the image below from Wikipedia:

[![enter image description here](https://i.stack.imgur.com/xiHhk.png)](https://i.stack.imgur.com/xiHhk.png)

Does this mean that if you add torsion, and you have an elevator falling through that geodesic, then the elevator will rotate together with the system of coordinates? That is, that the observer's rods and clocks are all rotating together along the geodesic? (if this is not the case, then what would be an accurate description of what the above figure means?)

If this were true, would not that object's motion look different in the case in which I experience torsion versus in a case in which I don't? If I am rotating, wouldn't I see an object moving on a straight geodesic far from me as revolving around me (specially the anything on the cosmic horizon)? So, wouldn't torsion affect the motion of non-local objects?

Answer: 
The answer to your first question is yes. The math for the special case of non-zero torsion on a manifold with a flat Cartesian metric causing rotation in a left-handed manner can be found at the bottom of p5 of the 18p paper by Steuard Jensen, titled General Relativity with Torsion: Extending Wald's Chapter on Curvature. The paper also references "Einstein-Cartan theory" a 1922 extension to GR showing that differential geometry is equally well-defined with torsion as without. Jensen's paper is a terrific go-to for all things torsion. Your other questions are tougher since the geodesic with torsion has dependency on the character of the metric tensor as well as the torsion tensor.
Your first statement is true in special cases but not in the general case (for example, a non-symmetric metric). Your second statement about basis vectors being independent of a connection is dubious since the metric is, by definition, the pair-wise dot products of the basis vectors. Remember that in order to even have a geodesic, we need a manifold with structure such as topology, charts, atlas and importantly, a connection (a way to take a covariant derivative). You cannot say the geodesics are different you can only say the geodesics are what they are. Think of torsion as rotating the entire tangent space. Full disclosure - I am very new at this subject which seems bottomless. Something else that might help you would be to look into the Lie algebra pertaining to torsion.