This is a special case of an involutive homography of a conic (see InvolutiveHomography.html for the general case). It is a map T of the conic onto itself defined by fixing a direction (line) d. To every point X of the conic X'=T(X) is the other intersection point of the line d' parallel to d with the conic.
T extends to a homography of the whole projective plane. For this consider the level conics generated by c i.e. when c is given by a quadratic equation f(x,y) = k consider the conics c' resulting by varying k. T can be extended to act on every conic by the same recipe as in c. Notice that all conics f(x,y)=k share the same principal axes.
The homography axis in this case is the conjugate to direction d diameter d' of the conic.