When a sounding rocket rolls about its longitudinal axis, the role of angle of attack and sideslip angle exchange regularly. This interchange causes the aerodynamic moments to alternate between pitching and yawing moments, which, coupled with the moment of inertia effects, further aggravates the inertial cross-coupling. In this paper, first we analyze the cross-coupling phenomena and derive a Linear, Time-Invariant (LTI), multi input - multi output model of a spinning sounding rocket and then we design an autopilot for this system. An application of multivariable control technique is presented to the design of the autopilot. The involved controller is advanced by combining Eigenstructure Assignment (EA) approach with the Particle Swarm Optimization (PSO) algorithm. Results of linear and nonlinear simulations are reported to demonstrate the performance and stability margin of the considered autopilot.