Persistent claims have been made over the last ~15yr that extragalactic redshifts, when corrected for the Sun's motion around the Galactic centre, occur in multiples of ~24 or ~36km/s. A recent investigation by us of 40 spiral galaxies out to 1000km/s, with accurately measured redshifts, gave evidence of a periodicity ~37.2-37.7km/s. Here we extend our enquiry out to the edge of the Local Super cluster (~2600km/s), applying a simple and robust procedure to a total of 97 accurately determined redshifts. We find that, when corrected for related vectors close to recent estimates of the Sun's galactocentric motion, the redshifts of spirals are strongly periodic (P~37.6km/s). The formal confidence level of the result is extremely high, and the signal is seen independently with different radio telescopes. We also examine a further sample of 117 spirals observed with the 300-foot Green Bank telescope alone. The periodicity phenomenon appears strongest for the galaxies linked by group membership, but phase coherence probably holds over large regions of the Local Supercluster.