We use HRTS-VI rocket observations of the solar chromosphere to search for relationships between high-Dopplershift ``jets'' observed in the C I lines near λ = 156 nm and internetwork ``bright points'' observed in the λ = 160 nm continuum, in sequel to the analysis by Cook et al. which failed to find a direct connection between these phenomena. We now use the same data to establish statistical correlations between C I Dopplershift and 160 nm brightness modulation in internetwork areas. These mean relations emerge only after extensive spatial averaging and have small amplitude, but are definitely significant. They show that both C I Dopplershift and 160 nm brightness participate in oscillatory behavior with 3 minute periodicity and mesoscale (8 Mm wavelength) as well as small-scale (1.4 Mm wavelength) spatial patterning. We find spatial and temporal phase relations between Dopplershift and brightness that confirm that jets and bright points should not be interpreted as isolated entities. Rather, they are chromospheric manifestations, with much pattern interference, of the oscillatory acoustic shock dynamics in the internetwork which also cause Ca II K2V grains. Additional small-scale modulation is present which we attribute to waves with f-mode character.