We present a clustering analysis of Luminous Red Galaxies (LRGs) using nearly 9 000 objects from the final catalogue of the 2dF-SDSS LRG And QSO (2SLAQ) Survey. We measure the redshift-space two-point correlation function, xi(s), at the mean LRG redshift of z=0.55. A single power-law fits the deprojected correlation function, xi(r), with a correlation length of r_0=7.45+-0.35 Mpc and a power-law slope of gamma=1.72+-0.06 in the 0.4<r<50 Mpc range. But it is in the LRG angular correlation function that the strongest evidence for non-power-law features is found where a slope of gamma=-2.17+-0.07 is seen at 1<r<10 Mpc with a flatter gamma=-1.67+-0.03 slope apparent at r<~1 Mpc scales. We use the simple power-law fit to the galaxy xi(r) to model the redshift space distortions in the 2-D redshift-space correlation function, xi(sigma,pi). We fit for the LRG velocity dispersion, w_z, Omega_m and beta, where beta=Omega_m^0.6/b and b is the linear bias parameter. We find values of w_z=330kms^-1, Omega_m= 0.10+0.35-0.10 and beta=0.40+-0.05. These high redshift results, which incorporate the Alcock-Paczynski effect and the effects of dynamical infall, start to break the degeneracy between Omega_m and beta found in low-redshift galaxy surveys. This degeneracy is further broken by introducing an additional external constraint, the value of beta(z=0.1)=0.45 from 2dFGRS, and then considering the evolution of clustering from z~0 to z_LRG~0.55. With these combined methods we find Omega_m(z=0)=0.30+-0.15 and beta(z=0.55)=0.45+-0.05. Assuming these values, we find a value for b(z=0.55)=1.66+-0.35. We show that this is consistent with a simple ``high peaks bias prescription which assumes that LRGs have a constant co-moving density and their clustering evolves purely under gravity. [ABRIDGED]
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