We have examined the orientations of early-type galaxies in the Coma cluster to see whether the well-established tendency for brightest cluster galaxies to share the same major axis orientation as their host cluster also extends to the rest of the galaxy population. We find no evidence of any preferential orientations of galaxies within Coma or its surroundings. The implications of this result for theories of the formation of clusters and galaxies (particularly the first-ranked members) are discussed.