  We consider top-antitop quark $(t{\bar t})$ production at the Large Hadron
Collider (LHC) with subsequent decays into dileptonic final states. We use and
investigate a set of leptonic angular correlations and distributions with which
all the independent coefficient functions of the top-spin dependent parts of
the $t{\bar t}$ production spin density matrices can be experimentally probed.
We compute these observables for the LHC center-of-mass energy 13.6 TeV within
the Standard Model at next-to-leading order in the QCD coupling including the
mixed QCD-weak corrections. We determine also the $t{\bar t}$ charge asymmetry
where we take in addition also the mixed QCD-QED corrections into account. In
addition we analyze and compute possible new physics (NP) effects on these
observables in terms of a gauge-invariant effective Lagrangian that contains
the operators up to mass dimension six that are relevant for hadronic $(t{\bar
t})$ production. First we compute our observables inclusive in phase space. In
order to investigate which region in phase space has, for a specific
observable, a high NP sensitivity, we determine our observables also in
two-dimensional $(M_{t{\bar t}},\cos\theta_t^*)$ bins, where $M_{t{\bar t}}$
denotes the $t{\bar t}$ invariant mass and $\theta_t^*$ is the top-quark
scattering angle in the $t{\bar t}$ zero-momentum frame.
