In this paper, a geometric approach is proposed for the design of minimum-time Eigen-axis repointing maneuvers of symmetric spacecraft equipped with four reaction wheels in a pyramidal configuration. First, by analyzing the geometric distribution of the available control torque, an algorithm is developed to compute the maximum achievable torque along an arbitrary direction. Based on this maximum torque, the constant control coefficients required for implementing a bang–bang control law in the Eigen-axis maneuver are determined. Analytical and numerical results demonstrate that, due to the non-uniform torque distribution inherent in the pyramidal configuration, selecting an Eigen-axis perpendicular to the plane defined by the initial and final pointing directions does not necessarily yield the minimum maneuver time. Instead, the optimal Eigen-axis is located at an angle smaller than 90 degrees with respect to this plane. Furthermore, the torque distribution and maneuver performance are investigated under the condition of failure of one reaction wheel and compared with the conventional three orthogonal reaction wheel configuration. Numerical simulations show that the pyramidal configuration provides superior maneuver performance in terms of achievable torque and maneuver time, even in the presence of a single wheel failure.