A computational study will be presented of the reflection of an inward--facing conical shock wave from its axis of symmetry. This is related to more complex practical situations in both steady and unsteady flows. The absence of a length scale in this problem makes features grow linearly with time. The ensuing flow is related to the Guderley singularity of a cylindrical imploding shock. The problem was explored by making a large number of computations of the Euler equations. Distinct reflection configurations, different from those encountered in shock reflection from a plane of symmetry, were identified, and the regions of parameter space in which they occur could be delineated.