The shot "donuts" because the whole shot cup is imparted with a spin, similar to the rotation on the bullet. When the shot cup clears the end of the barrel, the shot is spun out in all directions (except straight).
The effect was less pronounced in the age before shot cups but, even then, patterns were negatively affected. In the absence of a shot cup, the shot on the outter edge of the shot column gets flat-sided by the rifling grooves. The entire shot column spins and the shot is sprayed out in a vortex as it leaves the barrel. The difference between with and without a shot cup is that, without the shot cup, there are "fliers" caused by spinning the shot column and other fliers caused by the poor aerodynamics of flat-sided shot. Another difference is that, without a shot cup, a few pellets might actually hit the target.