The tail is the primary distinction between a double tail betta and a halfmoon betta. A double tail betta has a genetic mutation causing it to have two tails.  A halfmoon betta, on the other hand, has only one tail. Its tail is in the shape of a half-moon, giving it its name.   There's also a betta breed known as the double tail halfmoon. This fish comes from breeding a double tail betta with a halfmoon betta. It results in bettas with two half-moon-like tails.