Think I may have figured out the source of my confusion. Please tell me if this makes sense.
Since the Jacobian is taken for some scalar-valued or vector-valued function with respect to another vector, we're actually considering the FUNCTION which takes one coordinate system to another.
So when I want to change an integral from Cartesian coordinates to cylindrical coordinates I have to use the following functions:
x = g( r, theta, z) = rcos(theta)
y = h(r, theta, z) = rsin(theta)
z = m(r, theta, z) = z
The functions g, h, m when aggregated into a vector are a map FROM (r, theta, z) TO (x,y,z). So even though I use said map to change my integral from Cartesian to cylindrical, the map itself is from cylindrical coordinates into (x,y,z) coordinates.
I think that's why everything seemed "reversed" to me. Have I got it now, though?