  The short gravitational wave signal from the merger of compact binaries
encodes a surprising amount of information about the strong-field dynamics of
merger into frequencies accessible to ground-based interferometers. In this
paper we describe a previously-unknown "precession" of the peak emission
direction with time, both before and after the merger, about the total angular
momentum direction. We demonstrate the gravitational wave polarization encodes
the orientation of this direction to the line of sight. We argue the effects of
polarization can be estimated nonparametrically, directly from the
gravitational wave signal as seen along one line of sight, as a slowly-varying
feature on top of a rapidly-varying carrier. After merger, our results can be
interpreted as a coherent excitation of quasinormal modes of different angular
orders, a superposition which naturally "precesses" and modulates the
line-of-sight amplitude. Recent analytic calculations have arrived at a similar
geometric interpretation. We suspect the line-of-sight polarization content
will be a convenient observable with which to define new high-precision tests
of general relativity using gravitational waves. Additionally, as the nonlinear
merger process seeds the initial coherent perturbation, we speculate the
amplitude of this effect provides a new probe of the strong-field dynamics
during merger. To demonstrate the ubiquity of the effects we describe, we
summarize the post-merger evolution of 104 generic precessing binary mergers.
Finally, we provide estimates for the detectable impacts of precession on the
waveforms from high-mass sources. These expressions may identify new precessing
binary parameters whose waveforms are dissimilar from the existing sample.
