The elements of a matrix may form any shape if one wisely select elements. The given can not be called a double-cone matrix as it has axis elements as zeros and both parts are disjoint. You should take at least axis elements.
Konrad I think you are asking this question to look up theory on this type of matrix. I do not know of any special name for this matrix, but one thing you can do is email Horn of Johnson and Horn's Matrix Analysis and ask if he knows of any literature on these types of matrices. My inclination is to transform the matrix into another matrix that is more standard if possible. It seems like you can get something nice from a transformation. Hope this helps and would be interested if you find the answer.