Measurements from the CMS experiment at the LHC of dihadron correlations for charged particles produced in PbPb collisions at a nucleon-nucleon centre-of-mass energy of 2.76 TeV are presented. The results are reported as a function of the particle transverse momenta (p(T)) and collision centrality over a broad range in relative pseudorapidity (Delta eta) and the full range of relative azimuthal angle (Delta phi). The observed two-dimensional correlation structure in Delta eta and Delta phi is characterised by a narrow peak at (Delta eta, Delta phi) approximate to (0, 0) from jet-like correlations and a long-range structure that persists up to at least vertical bar Delta eta vertical bar = 4. An enhancement of the magnitude of the short-range jet peak is observed with increasing centrality, especially for particles of p(T) around 1-2 GeV/c. The long-range azimuthal dihadron correlations are extensively studied using a Fourier decomposition analysis. The extracted Fourier coefficients are found to factorise into a product of single-particle azimuthal anisotropies up to p(T) approximate to 3-3.5 GeV/c for at least one particle from each pair, except for the second-order harmonics in the most central PbPb events. Various orders of the single-particle azimuthal anisotropy harmonics are extracted for associated particle p(T) of 1-3 GeV/c, as a function of the trigger particle p(T) up to 20 GeV/c and over the full centrality range.