I posted this question on math.stackexchange but got no answer, so I decided to post it here instead. Sorry about the impreciness, not professional mathematician here.

Let's assume we have a set of 2D-points, and their coordinates are in a $n \times 2$ matrix. My claim is that if that set has at least one valid symmetry axis, then at least one of those axises is equivalent to one of the two principal component vectors, ie. eigenvectors of the covariance matrix.

For example, let's assume we have points (2,0), (-2,0), (0,1) and (0,-1). The eigenvectors are (1,0) and (0,1) (times some non-zero real number). In this case, those vectors are also the symmetry axises of the set of points.

Then again, take points (0,1), (0,-1), (5,2), (5,-2). Now vector (1,0) is a symmetry axis but (0,1) isn't.

In both cases at least one eigenvector is also a symmetry axis. Is this always the case, and if it is, how could I prove it?