  We consider real symmetric and complex Hermitian random matrices with the
additional symmetry $h_{xy}=h_{N-x,N-y}$. The matrix elements are independent
(up to the fourfold symmetry) and not necessarily identically distributed. This
ensemble naturally arises as the Fourier transform of a Gaussian orthogonal
ensemble (GOE). It also occurs as the flip matrix model - an approximation of
the two-dimensional Anderson model at small disorder. We show that the density
of states converges to the Wigner semicircle law despite the new symmetry type.
We also prove the local version of the semicircle law on the optimal scale.
