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 (Δη) and the full range of relative azimuthal angle (Δϕ). The observed two-dimensional correlation structure in Δη and Δϕ is characterised by a narrow peak at (Δη,Δϕ)≈(0,0) from jet-like correlations and a long-range structure that persists up to at least |Δη|=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≈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.