In this paper, we present a practical approach for the characterization of critical points on conical intersection seams as either local minima or saddle points using second-derivative technology. The utility of this methodology is illustrated by the analysis of seven S-0/S-1 (2A(g)/1A(g)) conical intersection points involved in the photochemistry of butadiene. The characterization of critical points on the crossing seam requires second derivatives computed in curvilinear coordinates. Using such coordinates, we can represent the branching space and the intersection space to second order. Although these curvilinear coordinates are conceptually important, they also give rise to two additional practical applications. First, such coordinates yield information on the nature of vibrational modes that are stimulated following radiationless decay at a crossing point. Second, the second-order force field is directly comparable to experimental spectroscopic data for Jahn-Teller systems. We will illustrate the latter idea for the cyclopentadienyl radical.