As Sousa mentioned below, the Sousaphone was essentially an adapted Helicon (a coiled bass instrument first produced in Vienna in 1845, the name coming from the Greek word for "the mountain of the Muses" - apparently being only coincidentally related to the word helix, meaning "a coil"). You can see the similarities here, with a Helicon on the left and a Sousaphone on the right (both in BBb; from a Conn advertisment in 1907):
The primary difference is the bell size and direction. The Helicon bell is considerably smaller, and points somewhat forward and to the left of the player (and it is not detachable), enabling the instrument to be carried easily by the player, whether he was on foot or on horseback (yep, they rode with these instruments in cavalry bands!). But it also created the tone that was, according to Sousa, "apt to shoot ahead too prominently and explosively to suit me for concert performances."
Above is a Helicon imported for J. W. Pepper in the 1880s, and below is that very first Sousaphone built by Pepper following Sousa's request in 1892. And so begins the life of this unusual member of the tuba family.
UPDATE: For more about the history of the helicon - especially in Sousa's Band - click here.