I have an oriented surface S and an oriented curve C. The surface and the curve intersect in points A and B. In A, the orientation of the curve is the same of the surface, in B they are in opposite directions. 
There are two vector fields, $v$ and $u$. These vector fields are independent in every point. On S, they constitute the generators of the tangent space of S. I think that there must be a point along C in which the vector tangent to the curve is a linear combination of $u$ and $v$.
I have an intuitive and non-rigorous reason. The three vectors $v$, $u$, and $n$ (a vector giving the orientation of $S$) have the same orientation on the whole surface S: they are either right-handed or left-handed. Moving along C, I consider $v$, $u$, and $m$, where $m$ is the vector tangent to the curve. Then these three vectors must change from left-handed to right-handed (or vice-versa). In the point where the change takes place, $m$ must be in the plane generated by $u$ and $v$.
Is this true? In case, how can we prove this? Does this property have a name?