Recognitions:

torsion in two dimensions?

Tricky, I think you're talking about something else. You seem to be referring to the torsion of curves, which is an extrinsic curvature that measures the rate at which the curve leaves the osculating plane.

What Ben asked about in the OP is the torsion of a connection, which is an intrinsic curvature that measures twisting of frames under parallel transport.

These are both called "torsion", but they are completely different things.

 Quote by Ben Niehoff Tricky, I think you're talking about something else. You seem to be referring to the torsion of curves, which is an extrinsic curvature that measures the rate at which the curve leaves the osculating plane. What Ben asked about in the OP is the torsion of a connection, which is an intrinsic curvature that measures twisting of frames under parallel transport. These are both called "torsion", but they are completely different things.
As I said in surfaces the connection skew-symmetric part that constitutes the torsion tensor is determined by only two independent components, the one-form(torsion form) in 2 dimensions. And usually vectors and covectors are not considered invariants in the same way scalars and tensors with more than one index are.

Is there something specifically that you disagree with in the previous paragraph?
 Looking at section 5.8.2 "The torsion tensor", of, http://www.lightandmatter.com/html_b...tml#Section5.8 it looks like the answer is kind of. From the above, "Torsion that does not preserve tangent vectors will have nonvanishing elements such as τxxy, meaning that parallel-transporting a vector along the x axis can change its x component. Torsion that preserves tangent vectors will have vanishing τλμν unless λ, μ, and ν are all distinct." If you include time as a dimension does that allow for torsion with 2 space dimensions? Edit, I just noticed I referenced bcrowell's work, %^). Its an interesting read, thank you for it!