Hadseaxes are a specific type of seax.
Hadseaxes were popular in the Continent into the eleventh century. At around 18” long, they combine the benefits and shortcomings of both a short seax and a langseax. Frequently having sharp points, they were useful for punching through maille to finish off a fallen opponent.
The typology of seax shapes (including Frankish examples) can be found on the general seax page.
Hadseaxes should really be restricted to Norman and Frankish warriors. They may be a primary or secondary combat weapon for lower-status Normans. Higher-status Normans are usually shown carrying a spear or sword, with no seax visible. While it is possible that a hadseax is concealed under a mail shirt, it seems unlikely and uncomfortable. It would be more reasonable to expect that opponents injured by a miles would be captured or finished off by more lightly-armed troops following behind.
It may not be practical to wear a hadseax horizontally across the belt. Instead, it can be slung from the belt at around 45 degrees, from one or more suspenders, in the same way as a sword or langseax.
See here for further information on seax sheaths. Note that seax sheaths should always be made with a sharp point at the end, even if the actual weapon is a combat blunt with a rounded end.