Correlations between mean transverse momentum [p_T] and anisotropic flow coefficients v₂ or v₃ are measured as a function of centrality in Pb–Pb and Xe–Xe collisions at s_NN=5.02 TeV and 5.44 TeV, respectively, with ALICE. In addition, the recently proposed higher-order correlation between [p_T], v₂, and v₃ is measured for the first time, which shows an anticorrelation for the presented centrality ranges. These measurements are compared with hydrodynamic calculations using IP-Glasma and T_RENTo initial-state shapes, the former based on the Color Glass Condensate effective theory with gluon saturation, and the latter a parameterized model with nucleons as the relevant degrees of freedom. The data are better described by the IP-Glasma rather than the T_RENTo based calculations. In particular, Trajectum and JETSCAPE predictions, both based on the T_RENTo initial state model but with different parameter settings, fail to describe the measurements. As the correlations between [p_T] and v_n are mainly driven by the correlations of the size and the shape of the system in the initial state, these new studies pave a novel way to characterize the initial state and help pin down the uncertainty of the extracted properties of the quark–gluon plasma recreated in relativistic heavy-ion collisions.