Picture of the Indiscriminate Interval #000008 - Eurhinodelphis longirostris

Eurhinodelphis longirostris at the American Museum of Natural History.

The most striking trait of Eurhinodelphidae is a toothless extension of the rostrum beyond the mandible (Lambert 2005), superficially similar to the bills of Billfish and Swordfish. Oddly, this morphology was speculative for a period of time (Kellogg 1925) although it has apparently been confirmed in several species as of Lambert (2005). Unfortunately, information on eurhinodelphids is scant and/or difficult to access and, among numerous other basics of their biology, I really don't know what the function of the extended rostrum would be. The only suggestion I could find is from one professor Abel who speculated that the cetaceans "swam on the surface of the sea, where they captured food - probably fishes - in much the same manner as does the skimmer (Rhynchops) [sic] among birds" (Anonymous 1909). Somehow, I find this even less plausible than azhdarchids-as-skimmers. On a curious note, there is a cetacean with the reverse of eurhinodelphid morphology (mandible extending past rostrum) unofficially known as the... skimmer porpoise.

Phylogenetically, eurhinodelphids have bounced around from being considered stem-ziphiids, the sister group to Delphinida, and the sister group to Squalodontidae + Squalodelphidae (Geisler et al. 2011 - citing various); within Geisler et al. (2011), they were placed outside crown-Odontoceti¹ in an unconstrained analysis and as the sister group of platanistoids in a constrained analysis. The authors regarded the latter position as more probable and placed eurhinodelphids within the new group Synrhina, which includes most odontocetes except for Sperm Whales and assorted extinct taxa. Whatever their placement, eurhinodelphids are certainly close relatives of living toothed whales, despite that whole extinct thing.

¹ It actually states they "did not fall inside crown Cetacea", but this is a typo. Otherwise, they'd be Miocene Archaeocetes. 

Eurhinodelphis longirostris seems to have an unusually long neck for a cetacean. The cervical vertebrae are not fused (Kellogg 1925), however this is a surprisingly common trait shared with river dolphins, monodontids, rorquals, and gray whales (Tinker 1988). The neck of E. longirostris appears to be proportionally longer than those of the baleen whales and Narwhal and is probably comparable to those of the Beluga and Dorudon. River dolphin skeletons are hard to find, but it seems likely they have similarly proportioned necks. It seems that Eurhinodelphis wasn't a total freak, well, except for the snout.

The Theatrical Tanystropheus covered Eurhinodelphis as well, and it doesn't even overlap that much!

References:

Anonymous. (1909). Notes. Nature 2088 (82), 16. Available.

Geisler, J. H., McGowen, M. R., Yang, G., Gatesy, J. (2011). A supermatrix analysis of genomic, morphological, and paleontological data from crown Cetacea. BMC Evolutionary Biology 11 (112). Available.

Kellogg, R. (1925). On the occurrence of fossil porpoises of the genus Eurhinodelphis in North America. Proceedings of the U. S. National Museum 66(26), 1-40. Available.

Lambert, O. (2005). Les dauphins longirostres et les baleines à bec du Néogène de la région d’Anvers: systématique, phylogénie, paléo-écologie et paléo-biogéographie. Doctoral Thesis. Partially Available.

Tinker, S. W. (1988). Whales of the World. E. J. Brill Publishing Company: New York. Partially Available.