We report on detailed small-angle neutron scattering measurements with polarization analysis from the flux-line lattice in the anisotropic superconductor YBa2Cu3O7-δ. When the field was applied at an angle to the principal axes we have observed spin-flip neutron diffraction consistent with local transverse field components. From a detailed study of the angle dependence of the magnitude of the spin-flip neutron-scattered intensity we have found that the transverse-field components are larger than predicted by an anisotropic London model for reasonable mass anisotropy values. The transverse-field components are consistent with a reduced c-axis coherence length at low temperature.