This paper proposes a novel, reconfigurable parallel kinematics machine with three degrees of freedom that can be used for various three-axis manipulation tasks, including machining. By locking some joints, the proposed parallel kinematics machine (PKM) can be transformed into four topologies with eight configurations to attain certain kinematic properties while keeping the number of its degrees of freedom unchanged. Either the proximal or intermediate prismatic joints of the reconfigurable PKM can be actuated. Some of the configurations are orthogonal configurations having a large rectangular cuboid workspace, and some other configurations are non-orthogonal configurations which provide the capability to perform a machining task to a large workpiece in various positions with respect to the machine. Accordingly, the proposed machine can be transformed from an orthogonal machine to a non-orthogonal machine with the advantages of each. The mobility of the various topologies of the reconfigurable PKM is rigorously analyzed using the screw theory. The workspace is analyzed using a graphical approach and verified by a computational approach. The pose kinematics shows that the various topologies have unified kinematics. The differential kinematics shows that the singularities in the various configurations occur at the workspace boundary. Similarly, the stiffness analysis shows that the low-stiffness postures occur around the workspace boundary. Accordingly, a used workspace far from the workspace boundary easily avoids the singularities and the low stiffness.