Local configuration controllability and local equilibrium controllability of the Triaxial attitude control testbed (TACT) are studied. The TACT is controlled only by a class of actuators referred to as shape actuators. This implies that there is a conserved quantity and there is a base body equilibrium manifold at each fixed shape. Thus, the TACT states are not small time locally controllable at equilibrium. But a weaker configuration controllability and equilibrium controllability property may be satisfied at an equilibrium. Important symmetric product formulas and properties are obtained. These properties explicitly show dynamic coupling and symmetry of the TACT. Based on these properties, local equilibrium controllability analysis is carried out for the TACT with fully actuated shape variables. Sufficient conditions for local equilibrium controllability are presented for three examples.