We examine the problem of agile trajectory tracking for a quadrotor vehicle with bidirectional thrust capabilities. A time-dependent linearization of the dynamics is presented based on the Lie algebra of $SO(3)$. The obtained linear dynamics are used as the basis for a nonlinear model predictive control algorithm that does not require operating around equilibrium state-input pairs. Our approach allows the problem to be transcribed as a quadratic program with an analytical Hessian. A method for ensuring the controller satisfies the nonlinear motor constraints involved with bidirectional thrust is then suggested, and the modelling of the thrust dynamics is discussed. Simulations for the performance of the controller are carried out on a basic half-flip maneuver as well as challenging aggressive turnaround and barrel roll maneuvers. The proposed controller performs well on all three maneuvers, significantly outperforming a reference Euler angle-based nonlinear MPC formulation and improving upon the simplified handling of the thrust constraints over the prediction horizon of the MPC.