Correlations between the elliptic or triangular flow coefficients vm (m = 2 or 3) and other flow harmonics vn (n = 2 to 5) are measured using √sNN = 2.76 TeV Pb + Pb collision data collected in 2010 by the ATLAS experiment at the LHC, corresponding to an integrated luminosity of 7 μb−1. The vm-vn correlations are measured in midrapidity as a function of centrality, and, for events within the same centrality interval, as a function of event ellipticity or triangularity defined in a forward rapidity region. For events within the same centrality interval, v3 is found to be anticorrelated with v2 and this anticorrelation is consistent with similar anticorrelations between the corresponding eccentricities, ∊2 and ∊3. However, it is observed that v4 increases strongly with v2, and v5 increases strongly with both v2 and v3. The trend and strength of the vm-vn correlations for n = 4 and 5 are found to disagree with ∊m-∊n correlations predicted by initial-geometry models. Instead, these correlations are found to be consistent with the combined effects of a linear contribution to vn and a nonlinear term that is a function of v22 or of v2v3, as predicted by hydrodynamic models. A simple two-component fit is used to separate these two contributions. The extracted linear and nonlinear contributions to v4 and v5 are found to be consistent with previously measured event-plane correlations.