Visualizing why the determinant changes sign on column swap

If a matrix is broken into columns, the determinant of a matrix can be visualized as the volume enclosed by those vectors. Ask A Mathematician/Ask a Physicist had a very nice post explaining this. However, it was not clear to me why the volume changes signs when you interchange the vectors. It seems to me that:

Parallelepiped with one set of labels (image from the Ask A Mathematician article)

Is the same parallelepiped as:

Changing the labels doesn't change the parallelepiped

So, they have the same volume. But they do not have the same determinant. So, something is lacking in my intuition about determinants and volumes.

As a first step to understanding this aspect of determinants, I worked out the proof that swapping two vectors negates the determinant for the specific example of the determinant of a two-dimensional matrix, keeping track of the geometric intuition at each step.

For a two dimensional matrix, the determinant is the signed area of the enclosed parallelogram.