Many of the types and styles of hatchets that existed at the time generally were designed for a specific function and task. Many of which found a carry over into soldiers hands that might not have been the original intended purpose. We find a wide array of types in period camp/troop photographs.
Some of the most commmon ones encountered..... Besides the regular ole "common hatchet"... We also find the "Half-Hatchet" (aka Carpenters Hatchet) that had a hammer head off the top edge of the other side. These continue to find popular use today, but most are slightly different than their period counterparts. This one was also closely related to what was known as a "Lathing Hatchet", that also had a hammer side but generally had a flat squared off top edge. its use was intended to work on wall wooden lathing strips that plaster was applied to. The "Shingling Hatchet"; normally had a centered hexigon sided hammer poll off the other side. A tool known as a "Fro" was used to spilt shingles, this tool was used for any required fine trimming and nailing on the shingles. This is also the same type that ole Lizzie Borden made famous.... The more hefty hatchet commonly found related to soldiers use is known as the "Broad Hatchet" aka "Hewing Hatchet". The blade was flat on one side and cutting edge only beveled on the other. This one was generally used for finer trimming/squaring work on a wide variety of wood items. Its bigger brother the huge wide bladed Broad Axe was also only beveled on one side, commonly used chip off the pre-scored sides to square off logs. (An "Adz" was also used for this purpose) the Broad/Hewing Hatchet was generally used for the smaller finer work. It was only beveled on one side so one could make a flat even cut trimming These are also commonly found in period photos of soldiers and wagon shops etc.
Many of the "types" saw extended use well after the war, Some changed very little if any at all... but some found slight design changes and improvements over the latter decades that might differ from their war-time cousins. Also noted that todays available replacement handle fodder, also may generally differ from those in common use at the time. Many can be reshaped to fit the bill a bit more accurately.