When do singular values occur in pairs?

Question

When playing around with a certain matrix I'm studying I noticed something interesting. The top singular values seem to occur in pairs:

In[1]:= N[SingularValueList[M]]
Out[1]= {24.955, 24.955, 13.9397, 13.9397, 9.83108, 9.83108, 7.73701,
7.73558, 6.53796, 6.51335, ...}

A plot of the whole spectrum looks like:

This seems special and not common. Are there certain classes of matrices for which this is known to occur?

Answer 1

It occurs in the presence of some symmetry, most obviously in matrices of the form

$$ \pmatrix{A & 0\cr 0 & A\cr}$$