The authors give the local classification of solution curves of bivalued direction fields determined by the equation \[ a(x,y)(dy)^ 2+2b(x,y)dy dx-a(x,y)(dx)^ 2=0, \] where a and b are smooth functions which vanish at \(0\in {\mathbb{R}}^ 2\). Such fields arise on surfaces in Euclidean space, near umbilics, as the principal direction fields, and also in applications of singularity theory to the structure of flow fields and monochromatic-electromagnetic radiation. A classification is given up to homeomorphism (there are three types) but the methods furnish much additional information concerning the fields, via a crucial blowing-up construction.