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We study a special class of (real or complex) robust Hadamard matrices, distinguished by the property that their projection onto a 2-dimensional subspace forms a Hadamard matrix. It is shown that such a matrix of order n exists, if there is a skew Hadamard matrix of a symmetric conference matrix of this size. This is the case for any even n ≤ 20, and for these dimensions we demonstrate that a bistochastic matrix B located at any ray of the Birkhoff polytope, (which joins the center of this bodydoi:10.1007/s11786-018-0384-y fatcat:4czcwgt4brebpfoodod2w5uwmi