In the Left-Right Symmetric Model (LRSM), box diagrams involving the charged right-handed gauge boson W_R may affect B-Bbar mixing as well as CP asymmetries in neutral B decays. The smallness of the epsilon_K parameter in the neutral K-meson system places severe constraints on the right-handed quark mixing matrix V^R, and reduces the number of its effective phases to one. W_R exchange gives a large contribution to B-Bbar mixing when the mass of the W_R boson is up to or higher than 8 TeV, depending on the V^R case, the B_d,s meson, and the asymmetry. The allowed regions of the CP violating asymmetries sin(2beta), sin(gamma), sin(2alpha), and sin(2phi_s), as well as x_s, are calculated as a function of the W_R mass. The results of the LRSM other than for the well measured sin(2beta) show allowable regions that are much broader than that for the Standard Model, showing that new experiments can indicate a presence of new physics, or significantly push up the limits on the W_R mass.