 Geodesic intersection from reflection symmetry

Suppose reflection in a plane is an isometry of a smooth surface, and its intersection with the plane is a regular curve. That curve is a geodesic when parametrized by arclength. At any intersection point, reflection fixes both the point and every tangent vector to the intersection. It carries the geodesic with these initial data to one with the same data. Local uniqueness of the geodesic equations forces it to remain in the fixed plane. This is the one-dimensional instance of fixed-point sets of Riemannian isometries being totally geodesic.