The coincidence of a bifurcation point with a snap-through point is called hilltop buckling. In this paper, it either serves as the starting point - the ? - or as the end - the O - in sensitivity analysis of the initial postbuckling behavior of elastic structures. It is shown that hilltop buckling is imperfection sensitive. In sensitivity analyses with hilltop buckling as the starting point (end), the bifurcation point and the snap-through point are diverging from (converging to) each other. Two classes of sensitivity analyses are identified by means of the consistently linearized eigenproblem. They determine the more (or less) effective mode of conversion of an originally imperfection-sensitive into an imperfection-insensitive structure. The results from the numerical investigation corroborate the theoretical findings. The present study is viewed as a step in the direction of better understanding the reasons for different modes of the initial postbuckling behavior of elastic structures and its interplay with the prebuckling behavior.