  Complex Hadamard matrices H of order 6 are characterized in a novel manner,
according to the presence/absence of order 2 Hadamard submatrices. It is shown
that if there exists one such submatrix, H is equivalent to a Hadamard matrix
where all the nine submatrices are Hadamard. The ensuing subset of
H_2-reducible complex Hadamard matrices is more general than might be thought,
and, significantly, includes all the up till now described (one- and
two-parameter) families of order 6. A known, isolated matrix, and most
numerically generated matrices, fall outside the subset.
