In this fourth paper of a series concerning charged particle behavior in ultralow frequency waves in the terrestrial magnetosphere, we examine the particle flux response expected in waves with a strong compressional magnetic component. Two effects, which we label betatron and mirror, dominate the behavior expected for nonresonant particles with the mirror effect expected in most circumstances. Resonant behavior is a strong function of signal symmetry, much as discussed in earlier papers. We conclude by examining recently published observations of particle flux oscillations associated with compressional signals.