The barrel roll in freefall: a roll around the long axis over your back that USPA and DFV use for stabilisation and that should end within 5 seconds.
The honest answer up front: the barrel roll is a roll around your body's long axis in which you turn over your back and come back to belly-to-earth. The USPA calls it "the best technique for returning to a belly-to-earth position" [1]. That is exactly why you learn it early: it is not show, it is a tool against instability.
This fits with how the associations use it. In the DFV manual, level VI includes a stabilisation exercise with a somersault or roll, performed down to 2,500 m above ground [2]. The USPA requires in category E that you finish the barrel roll within 5 seconds [1]. Whoever recovers from two such intentional instabilities (barrel roll, back loop or front loop) within 5 seconds may jump without direct freefall supervision [1].
The guide gives you the overview. It does not replace training. Practice happens with a coach or in the wind tunnel.
You lie relaxed, heading is there, altitude is right [1]. Milestone: you checked HAALR before the roll (Heading, Altitude, Arch, Legs, Relax).
The USPA describes: bring the knees together, stretch both arms forward over your head, keep your eyes on the horizon [1]. That makes you narrow and the roll easier.
You drive one arm across your chest like a punch and create momentum. With a scissoring action of arms and legs you twist around the axis from head to feet [1]. Milestone: you rotate evenly, not jerkily.
You return to the neutral position, relax and read the altimeter [1]. The USPA sets 5 seconds as the limit for recovery [1]. Milestone: after the roll you lie flat and know where the horizon is.
You repeat at sufficient altitude until the limit is reached [1]. Incidentally, the A-licence check requires a back loop within 60° of the starting heading [3]. Your coach shows you what he looks for.
This explains and motivates, it does not replace supervised training. Deliberately leaving the stable position is supervised teaching material.
The barrel roll is your way out, not your goal. If you end up on your back, the roll is the USPA's answer to how you get back to belly-to-earth [1]. So practise it in both directions (right and left) so that in an emergency you do not have to think about which side is "yours" (practical experience).
A roll around the long axis: you turn over your back and return to belly-to-earth. The USPA calls it the best technique for returning to belly-to-earth [1]. You start in the neutral position and end there.
The USPA requires in category E that you are recovered within 5 seconds [1]. Whoever recovers from two intentional instabilities within 5 seconds can be cleared for self-supervision [1].
In the USPA in category E after turns and leg work [1]. In the DFV manual level VI has a stabilisation exercise with somersault or roll, performed down to 2,500 m above ground [2].
It is a controlled part of training, but because of orientation and altitude display it is practised only under supervision. The USPA warns that an analogue altimeter in the burble may read high when inverted [1].
As recovery tools the USPA names HAALR (heading, altitude, arch, legs, relax) and the roll-out-of-bed technique, meaning a roll back to belly-to-earth [1]. More important than elegance is altitude: working time ends at the set limit, and in training you agree with your coach how to react there.
Back Loop: The Backward Somersault in Freefall