We consider the problem of classifying the lines of the projective 3-space PG(3,q) over a finite field Fq into orbits of the group PGL2(q) of linear symmetries of the twisted cubic C . The problem has been solved in literature in characteristic different from 3, and in this work, we solve the problem in characteristic 3. We reduce this problem to another problem, which is the classification of binary quartic forms into PGL2(q)-orbits. We first solve the latter problem and use to solve the former problem. We also obtain the point-line and the line-plane incidence structures of the point, line, and plane orbits.