I will report on joint work in progress with Aleksandar Milivojevic (MPIM Bonn) on the elementary topology of the space of almost complex structures on a manifold. First I will describe a certain natural parametrization and associated stratification of the space of linear complex structures on a vector space and give a lower bound for the number of complex k-planes jointly preserved by two linear complex structures. Then I will focus on dimension 6 and prove a formula for the homological intersection of two orthogonal almost complex structures on a Riemannian 6-manifold when these are regarded as sections of the twistor space.