Do both groups having this feature mean radial symmetry evolved twice?
Good question! I had never really though about it, so thank you!
Echinodermata have a pentaradial symmetry
Echinodermata actually don't have a radial symmetry like jellyfish do. They have a pentaradial symmetry as they systematically have 5 arms.
Even if Echinodermata were radially symmetric, then it would actually be unlikely that bilateral symmetry evolved, was then lost and evolved again. It is much more parsimonious that bilateral symmetry evolved once and was later lost in the echinodermata lineage.
Developmental and Genetics of pentaradial symmetry
As one would expect modifications of hox genes are involved into the evolution of this peculiar body plan. From Lowe and Wray 1997
Here we report the expression domains in echinoderms of three important developmental regulatory genes ( distal-less, engrailed and orthodenticle ), all of which encode transcription factors that contain a homeodomain. Our findings show that the reorganization of body architecture involved extensive changes in the deployment and roles of homeobox genes. These changes include modifications in the symmetry of expression domains and the evolution of several new developmental roles, as well as the loss of roles conserved between arthropods and chordates. Some of these modifications seem to have evolved very early in the history of echinoderms, whereas others probably evolved during the subsequent diversification of adult and larval morphology.
Here is the whole abstract of Mooi and David 2008
The strangeness of echinoderm pentaradiality results from superposition of radial symmetry onto ancestral deuterostome bilaterality. The Extraxial-Axial Theory shows that echinoderms also have an anterior/posterior (A/P) axis developed independently and ontogenetically before radiality. The A/P axis is first established via coelomic stacking in the extraxial region, with ensuing development of the pentamerous hydrocoel in the axial region. This is strongly correlated with a variety of gene expression patterns. The echinoid Hox cluster is disordered into two different sets of genes. During embryogenesis, members of the posterior class demonstrate temporal, spatial, and genetic colinearity within the extraxial region. We suggest that displacement of genes from the more anterior Hox classes toward the 5′ end of the chromosome leads to control of the later-developing, radially symmetric axial region. Genetic disorder is therefore another way of using colinearity to build the unique echinoderm symmetry.
As insinuated above, note that (in most echinodermata species at least) only adults have a radial symmetry. Larvea have a bilateral symmetry. Here are a bunch of picturesand schema.
It is actually hypothesized that radial symmetry did evolve again, but there are homologies between the echinoderm symmetry and bilateral animals: