Conulariids are scyphozoans characterised by their pyramidal shapes, which have been found in more or less straight to weakly curved forms. More strongly curved periderms are more often to be found in long individuals (~15 cm +), as happens with recent scyphozoans, e.g. the polyps of Atorella, that are normally attached to the underside or the flank surfaces of a host and develop upwards as they grow longer.
Reconstruction of living conulariids. Courtesy of Enrique Sendino.
Werner was the first researcher to compare conulariids to coronates and believed the first conulariids were ancestors of coronates. His theory has been echoed in numerous papers by different researchers for over 50 years.
Coronates and conulariids coincide in being marine solitary or colonial sessile organisms with tetramerous symmetry. Both have a high degree of regeneration capability, a reason why some conulariids have been found with scars, having being bitten and with the periderm later regenerated (see Figure below). They have been also found with the schott present. This happens when the apical part of the animal has been broken off as a consequence of the marine current, breaking at the sharpest point of the pyramidal shape and regenerating this apical part in a rounded wall.
Most of the conulariids we find nowadays in the sediment lack the apical part. In cases where the schott is retained at the lower end, it may have several transverse partitions. This has been observed in the largest specimens in the NHM collections. The probable explanation is that when conulariids became too large and heavy, they offered resistance to the water currents and only broke off from the thin base.
There is an NHM specimen that confirms this theory, a large specimen more than 15 cm long that is bent, very likely due to marine currents, and also has a schott. When the animal was alive and the apical part broken off, the periderm was being moved passively on the seafloor by the current, making the apical part rounded by erosion whilst at the same time the transverse schott was being constructed from the soft parts. In cases where the specimen has a sharp apical part, this signifies that the animal could not have survived for very long after it had drifted away from the sea floor.
‘Conularia’ pyramidata from Gres de May in Normandy (NHMUK PI CL 569). © The Trustees of the Natural History Museum, London, CC-BY 
Periderm with a circular scar, probably bitten by a shark, evidence of conulariid regeneration ability. Paraconularia ‘quadrisulcata’ (NHMUK PI CL 693), from the Carboniferous Limestone Supergroup of an unknown locality, probably in England
Conulariid periderms could remain attached to the sea floor even after the death of the organism. These periderms, although they are thin, were strong enough to protect the soft body from predators and at the same time could host other organisms such as bryozoans, echinoderms or brachiopods, but were also flexible and elastic. The remains found nowadays have different colours, from dark brown, reddish brown, violet brown to blue-grey, light grey and whitish. As they grew, new transverse ribs were added towards the aperture allowing its growth upwards. In consequence, the thickness of the periderm decreases from the apical part to the aperture.
By analogy with coronates, conulariids must have had long ribbon-like or thread-like, nematocyst-bearing, contractile feeding tentacles which left the periderm gastric space, passing through an aperture to feed the animal. The aperture of conulariids differs from the coronates. The conulariid aperture was closed by triangular flaps, while the coronates have a unique lid. The conulariid tentacles were used to attract prey with the help of the nettle cells, probably small plankton organisms by analogy with coronates. By contraction of the tentacles and the abrupt impact, the prey animals are transported whole into the stomach area. The indigestible remains are eliminated in the same way.
As Werner suggested, conulariids probably represent the most basic group of all Scyphozoa
To know more about this group read the following references:
Sendino, C. & Bochmann, M.M. 2021. An exceptionally preserved conulariid from Ordovician erratics of Northern European Lowlands. PalZ (Paläontologische Zeitschrift), 95, 71-84. https://doi.org/10.1007/s12542-020-00534-7.
Sendino, C., Broda, K. & Zaton, M. 2017. First record of true conulariids from the Upper Devonian of Poland. Proceedings of the Geologists’ Association 128, 401-406. DOI: 10.1016/j.pgeola.2017.03.004
Sendino, C. & Darrell, J. 2009. History of Conulariid Research. Journal of the Palaeontological Society of India, 54 (2), 121-133.
Sendino, C. & Darrell, J. 2009. The collection of conulariids of The Natural History Museum of London. The Geological Curator, 9 (1), 3-20.
Sendino, C. Taylor, P.D. & Van Iten, H. 2012. Metaconularia? pyramidata (d’Orbigny, 1850): a scyphozoan from the Ordovician of Normandy, France, recorded for the first time as a remanié fossil in the Triassic of Devon, England. Geodiversitas, 34 (2), 283-296.
Sendino, C., Zágorsek, K. & Vyhlasová, Z. 2011. The aperture and its closure in the Ordovician conulariid. Acta Palaeontologica Polonica, 56 (3): 659-663. http://dx.doi.org/10.4202/app.2010.0028
Sendino, C., Zágorsek, K. & Taylor, P. 2012. Asymmetry in an Ordovician conulariid cnidarian. Lethaia, 45, 423-431.