  We investigate a number of observables that are and will be instrumental in
the exploration of $\tbart$ production and decay at the Tevatron and the LHC.
For this analysis we made a computer program that incorporates besides the NLO
QCD corrections to $\tbart$ production and decay also mixed weak-QCD
corrections to the production amplitudes, and that allows for studies of
correlated versus uncorrelated $\tbart$ events. In this paper we analyze and
compute observables mostly for dileptonic $\tbart$ final states to
next-to-leading order in the strong and weak gauge couplings (NLOW), with
selection cuts. We calculate charge asymmetries of the top quark and of
$\ell=e,\mu$ and we compare, where possible, with experimental results. We show
that top-quark spin correlations affect a leptonic pair asymmetry (which has
not been measured so far) by $\sim 7%$. We determine several dileptonic angular
correlations, which reflect $\tbart$ spin correlations, namely the beam,
off-diagonal and helicity correlation, and the opening angle distribution
(defined in the $t$, $\bar t$ rest frames) when selection cuts are applied. Our
NLOW predictions for the beam, off-diagonal, and helicity correlation for the
Tevatron agree with recent measurements by the CDF and D0 experiments. In
addition we make predictions for estimators of these correlations as functions
of $\mtt$. These estimators may prove useful for the Tevatron and also in the
early rounds of LHC data-analyses, where the event numbers will probably not be
abundant. Furthermore, we compute to NLOW in the gauge couplings two dilepton
angular correlations that are defined in the laboratory frame, for correlated
and uncorrelated $\tbart$ events at the LHC. Finally, based on our SM results,
we identify several observables that allow to search for non-standard parity-
and CP-violating interactions, especially with future LHC data.
