Measurements of transverse energy-energy correlations and their associated azimuthal asymmetries in multijet events are presented. The analysis is performed using a data sample corresponding to 139 fb^-1 of proton-proton collisions at a centre-of-mass energy of root s = 13TeV, collected with the ATLAS detector at the Large Hadron Collider. The measurements are presented in bins of the scalar sum of the transverse momenta of the two leading jets and unfolded to particle level. They are then compared to next-to-next-to-leading-order perturbative QCD calculations for the first time, which feature a significant reduction in the theoretical uncertainties estimated using variations of the renormalisation and factorisation scales. The agreement between data and theory is good, thus providing a precision test of QCD at large momentum transfers Q. The strong coupling constant alpha(s) is extracted as a function of Q, showing a good agreement with the renormalisation group equation and with previous analyses. A simultaneous fit to all transverse energy-energy correlation distributions across different kinematic regions yields a value of alpha(s)( mZ) = 0.1175 +/- 0.0006 (exp.)(+0.0034) (-0.0017) (theo.), while the global fit to the asymmetry distributions yields alpha(s)(m(Z)) = 0.1185 +/- 0.0009 (exp.)(+0.0025)(-0.0012)(theo.).