If a linear transformation swaps two axes is said to perform a reflection and the determinant will be negative. Testing for the sign of the determinant can tell me whether a reflection has happened.
What if we swap two pairs of axes instead? I understand that each swap will change the sign of our determinant. So, does this mean the given linear transformation is no longer performing a reflection? I'm confused…
If you swap two pairs of axes, what you end up with is a $180^\circ$ rotation around another axis. For example, if we reflect the $y$ value and the $z$ value as follows, it's the same as a rotation by $\pi$. $$ \begin{align} \left( \begin{matrix} 1 & 0 & 0\\ 0 & -1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right) \times \left( \begin{matrix} 1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & -1 \\ \end{matrix} \right) &= \left( \begin{matrix} 1 & 0 & 0\\ 0 & -1 & 0 \\ 0 & 0 & -1 \\ \end{matrix} \right)\\ &= \left( \begin{matrix} 1 & 0 & 0\\ 0 & \cos\pi & \sin\pi \\ 0 & -\sin\pi & \cos\pi \\ \end{matrix} \right) \end{align} $$ The general result is that the composition of two reflections is a rotation. You can find various results and demonstrations by searching for that.
Clearly the determinant of the above example is positive, and the general result is that the composition of two reflections is not a reflection.