The orthothecids are one of the two hyolith orders.
Marek diagnoses the order thus: Conchs with a flat or concave ventral surface — opercula with large, flat cardinal processes but without clavicles – tightly sigmoidal, sediment-filled
intestine – helens absent.
Sometimes the Circothecidae and Tetrathecidae are split out into a separate order 'Circothecida', which is defined by the bottom surface not being flat, the cardinal processes being pronounced, and a circular rim sometimes showing hints of differentiation into clavicles.
Internal taxonomy
Marek gives the following diagnoses:
Orthothecidae: Flat or concave bottom surface of conical shell. Transverse aperture.
Kouchinsky lists the following taxonomic criteria:
Circothecidae: circular cross-section; same shape and sculpture on upper and lower surfaces.
Turcurthecidae: oval or lenticular cross-section; top and bottom identical
Allathecidae: top side flat or slightly convex; bottom side convex
↑Malinky, J.M.; Skovsted, C.B (2004). "Hyoliths and small shelly fossils from the Lower Cambrian of North−East Greenland". Acta Palaeontologica Polonica. 49 (4): 551–578.