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Matrices with rotational symmetry

I've seen a [claim without proof](https://mathoverflow.net/a/418547) that the characteristic polynomials of matrices with rotational symmetry (i.e. $n \times n$ matrices $A$ with $A_{i,j} = A_{n+1-i,n+1-j}$) always factor into the product of the characteristic polynomials of smaller matrices which can be derived from blocks of the original matrix. Is there an elementary proof, and can the result be generalised?