  Top-antitop pairs produced at hadron colliders are largely unpolarized, but
their spins are highly correlated. The structure of these correlations varies
significantly over top production phase space, allowing very detailed tests of
the Standard Model. Here, we explore top quark spin correlation measurement
from a general perspective, highlighting the role of azimuthal decay angles. By
taking differences and sums of these angles about the top-antitop production
axis, the presence of spin correlations can be seen as sinusoidal modulations
resulting from the interference of different helicity channels. At the LHC,
these modulations exhibit nontrivial evolution from near-threshold production
into the boosted regime, where they become sensitive to almost the entire QCD
correlation effect for centrally produced tops. We demonstrate that this form
of spin correlation measurement is very robust under full kinematic
reconstruction, and should already be observable with high significance using
the current LHC data set. We also illustrate some novel ways that new physics
can alter the azimuthal distributions. In particular, we estimate the power of
our proposed measurements in probing for anomalous color-dipole operators, as
well as for broad resonances with parity-violating couplings. Using these
methods, the 2012 run of the LHC may be capable of setting simultaneous limits
on the top quark's anomalous chromomagnetic and chromoelectric dipole moments
at the level of 3*10^{-18}cm (0.03/m_t).
