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This paper presents a geometric analysis and synthesis theory for quotient kinematics machines (QKMs). Given a desired motion type described by a subgroup G of the special Euclidean group SE(3), QKM refers to a left-and-right hand system that realizes G through the coordinated motion of two mechanism modules, one synthesizing a subgroup H of G, and the other a complement of H in G, denoted by G/H. In the past, QKMs were often categorized into hybrid kinematics machines (HKMs) and were treated on a case-by-case basis. We show that QKMs do have a unique and well-defined kinematic structure that permits a unified and systematic treatment of their synthesis and design. We also study the properties of G/H as a novel motion type for parallel kinematics machine (PKM) synthesis. Another contribution of the paper is to model five-axis machines by SE(3)/R(o,z) (where R(o,z) represents the spindle symmetry) and give a complete classification of five-axis QKMs using the same geometric framework.