  We develop various properties of symmetric generalized complex structures (in
connection with their holomorphic space and B-field transformations), which are
analogous to the well-known results of Gualtieri on skew-symmetric generalized
complex structures. Given a symmetric or skew-symmetric generalized complex
structure \mathcal J and a connection D on a manifold M, we construct an almost
complex structure J^{\mathcal J,D} on the cotangent manifold T^{*}M and we
study its integrability. For \mathcal J skew-symmetric, we relate the Courant
integrability of \mathcal J with the integrability of J^{\mathcal J, D}. We
consider in detail the case when M=G is a Lie group and \mathcal J , D are
left-invariant. We also show that our approach generalizes various well-known
results from special complex geometry.
