The quasienergy spectra of the generalized kicked particle are studied under quantum-resonance conditions for arbitrary quasimomentum. The latter quantity is most relevant in atom-optics experimental realizations of this paradigmatic nonintegrable system. For generic quasimomentum, a quasienergy spectrum is shown to exhibit the regular structure of a staggered ladder, i.e., it is the superposition of a finite number of ladder subspectra all having the same spacing, which is independent of the nonintegrability. Each subspectrum covers densely the entire quasienergy range. The main manifestation of this spectral structure is a dynamical localization whose frequency spectrum is a staggered ladder with the central ladder independent of the nonintegrability. A second manifestation are traveling-wave components in wave-packet evolution. These phenomena should be experimentally observable.