Jacobian for Conversion Between Polar and Cartesian Resembles a Rotation Matrix In matrix form, supposing $z=f(x,y)$ where x,y represent the cartesian coordinate system, one can write the chain rule:
$ \begin{bmatrix}
z_x \\
z_y 
\end{bmatrix} = \begin{bmatrix}
r_x & \theta_x \\
r_y & \theta_y 
\end{bmatrix}   
 \begin{bmatrix}
z_r \\
z_\theta 
\end{bmatrix}$
Which turns out to be
$ \begin{bmatrix}
z_x \\
z_y 
\end{bmatrix} =  \begin{bmatrix}
\cos\theta & -\frac{\sin\theta}{r} \\
\sin\theta & \frac{\cos\theta}{r} 
\end{bmatrix}  
 \begin{bmatrix}
z_r \\
z_\theta 
\end{bmatrix}$
Why does this middle matrix resemble a rotation and a scaling by 1/r so much?

Because that is what it is in a way.  The action of a rotation matrix is specified by the action of it on basis vectors. Under rotation of a given angle, e_x gets sent to r hat of that angle and e_y to theta hat of that angle.
