Why is cross product not commutative? Why, conceptually, is the cross product not commutative? Obviously I could simply take a look at the formula for computing cross product from vector components to prove this, but I'm interested in why it makes logical sense for the resultant vector to be either going in the negative or positive direction depending on the order of the cross operation. I don't have any formal experience in linear algebra, so I would appreciate if an answer would take that into account (I'm merely learning vectors as part of a 3D game math education).

The magnitude of the resulting vector is a function of the angle between the vectors you are multiplying. The key issue is that the angle between two vectors is always measured in the same direction (by convention, counterclockwise).
Try holding your left thumb and index finger in an L shape. Measuring counterclockwise, the angle between your thumb and index finger is roughly 90 degrees. If you  measure from your index finger to your thumb (still must be done counterclockwise!) you have roughly a 270 degree angle.

One way to calculate a cross product is to take the determinant of a matrix whose top row contains the component unit vectors, and the next two rows are the scalar components of each vector. Changing the order of multiplication is akin to interchanging the two bottom rows in this matrix. It is a theorem of linear algebra that interchanging rows results in multiplying the determinant by -1.
