Given a (2N - 1)-dimensional projective space over GF(2), PG(2N - 1, 2), and its geometric spread of lines, there exists a remarkable mapping of this space onto PG(N - 1, 4) where the lines of the spread correspond to the points and subspaces spanned by pairs of lines to the lines of PG(N - 1, 4). Under such mapping, a non-degenerate quadric surface of the former space has for its image a non-singular Hermitian variety in the latter space, this quadric being {\it hyperbolic} or {\it elliptic} in dependence on N being {\it even} or {\it odd}, respectively. We employ this property to show that generalized Pauli groups of N-qubits also form two distinct families according to the parity of N and to put the role of symmetric operators into a new perspective. The N=4 case is taken to illustrate the issue.