A reflection has eigenvalues which are either $-1$ and $1$. The dimensions of symmetry of reflection are the ones which are $1$ and the ones which are reflected are $-1$ Basically you can write them in this way:
$A = V^{-1}DV$ where $D$ is diagonal and the columns $V_{:,k}$ are the corresponding vectors which are either left alone or reflected (depending on if $D_{kk}$ is 1 or -1).
In three dimensions we just have 2 times as many combinations, each of the three values could be either 1 or -1, but the same principle holds.
- If one $-1$, then there is a plane which the vectors are reflected in.
- If two $-1$ then there is a "thread" or "uncooked spaghetti" of reflection around.
- If three $-1$ then each dimension is flipped 180 degrees.