In this article, two coupled patch radiators are intensively investigated to characterize two pairs of even/odd modes by virtue of characteristic modes analysis (CMA). It is discussed that both radiation patterns and characteristic angles (CAs) of the two orthogonal odd modes are critical factors in the design of dual-beam circularly polarized (CP) antennas. However, neither of them could be correctly derived from a two-element-array model. By adjusting resonant frequencies (ResFs) in different ways, the two odd modes could achieve a 90° difference in CA. Moreover, the two odd modes have similar dual-beam radiation patterns with peak values in the same vertical plane. By further effectively exciting the two odd modes and fully suppressing the two even ones, four CP dual-beam patch antennas are successfully designed. Both simulated and measured results validate the effectiveness of the proposed technique in the design of low-profile high-directivity dual-beam CP patch antennas. It is also discussed that the pointing angle of dual-beam could be reduced by enlarging patch distance or utilizing higher-order odd modes, while the axial-ratio (AR) bandwidth could be improved by increasing substrate height or introducing a broadband feeding network.