An accelerometer can detect changes in speed. It also detects gravitational and centrifugal force. The accelerometer can detect how fast it is swinging (proportional to the centrifugal force) and any changes in direction. The centrifugal force, and the rotation of the end of the poi means that "down" will almost always (depending on speed) be pointing outwards from the circle at the point where the poi is.
Gyroscopes detect angular momentum, or how fast / far they are being twisted. They are better at fixing the position of a rotating object than an accelerometer is. They are not affected by the centrifugal force, but cannot detect linear (or near linear) changes in velocity.
For this kind of motion you would really want both to be accurate.
And then, as has been mentioned, you have the maths. 3D trigonometry is not easy at the best of times. When you add time into the equation to make it 4D trigonometry your brain starts to ooze out of your ears.