Recently, it was shown that the complex dynamical behavior of spicules has to be interpreted as the result of simultaneous action of three kinds of motion: (1) field aligned flows, (2) swaying motions, and (3) torsional motions. We use high-quality observations from the CRisp Imaging SpectroPolarimeter at the Swedish 1-m Solar Telescope to investigate signs of these different kinetic modes in spicules on the disk. Earlier, rapid blue-shifted excursions (RBEs), short-lived absorption features in the blue wing of chromospheric spectral lines, were identified as the disk counterpart of type II spicules. Here we report the existence of similar absorption features in the red wing of the Ca II 8542 and Hα lines: rapid redshifted excursions (RREs). RREs are found over the whole solar disk and are located in the same regions as RBEs: in the vicinity of magnetic field concentrations. RREs have similar characteristics as RBEs: they have similar lengths, widths, lifetimes, and average Doppler velocity. The striking similarity of RREs to RBEs implies that RREs are a manifestation of the same physical phenomenon and that spicules harbor motions that can result in a net redshift when observed on-disk. We find that RREs are less abundant than RBEs: the RRE/RBE detection count ratio is about 0.52 at disk center and 0.74 near the limb. We interpret the higher number of RBEs and the decreased imbalance toward the limb as an indication that field-aligned upflows have a significant contribution to the net Dopplershift of the structure. Most RREs and RBEs are observed in isolation, but we find many examples of parallel and touching RRE/RBE pairs which appear to be part of the same spicule. We interpret the existence of these RRE/RBE pairs and the observation that many RREs and RBEs have varying Dopplershift along their width as signs that torsional motion is an important characteristic of spicules. The fact that most RBEs and RREs are observed in isolation agrees with the idea that transversal swaying motion is another important kinetic mode. We find examples of transitions from RRE to RBE and vice versa. These transitions sometimes appear to propagate along the structure with speeds between 18 and 108 km s -1 and can be interpreted as the sign of a transverse (Alfvénic) wave.