The aim of this paper is to introduce the use of a double Hooke's joint linkage to reconfigure the base revolute joints of a 4-RUU parallel manipulator (PM). It leads to an architecturally reconfigurable 4-rRUU PM whose platform motion depends on the angle between the driving and driven shafts of the double-Hooke's joint linkage in each limb. Even when the angle is fixed, the manipulator is reconfigurable by virtue of the different operation modes it can exhibit. By keeping the angle as a variable, the constraint equations of the 4-rRUU PM are derived using Study's kinematic mapping. Subsequently, the ideal of constraint polynomials is decomposed as an intersection of primary ideals to determine the operation modes of the 4-rRUU PM for intersecting and parallel revolute joint axes in the base and the moving platform.