When integrating differential forms, we know classically that
$$ dx \, dy = -dy \, dx$$
Which geometrically comes from the orientation of an area formed by the two forms, such that the counter-clockwise motion from $dx$ to $dy$ defines a positive area, and so forth.
Radians are typically given as being a dimensionless and unitless quantity, but observe the following integral expressed in polar coordinates:
$$ \int_\limits{\theta} \int_\limits{r} dr \, d\theta$$
(i) Does it hold still that $dr \, d\theta = - d\theta \, dr$? Furthermore, while in the cartesian coordinates $dx , dy$ we can see that walking in some direction has a vector-like quality to it, which extends to tangent bundles and makes integrals obvious.
(ii) If radians are truly dimensionless then how do you extend the wedge product to $d \theta$ and $dr$? Even neglecting integration and differentiation, are you not allowed to put a vector field on a surface given in polar coordinates? If either are true, it seems there must be a dimensional and oriented quantity associated with a displacement in the $\theta$-direction. Neglecting all formalism, I can intuitively walk counter-clockwise around a circle (positive radian displacement) and then walk clockwise instead (negative radian displacement).
What then would give radians any less legitimacy as a dimensional unit than meters or joules? Or perhaps the 360 degrees of the circle, where there's a similarity between the conversion of radians to degrees as there is for the conversion of Celsius to Fahrenheit.
Thanks in advance for the help.