Binary-black-hole dynamics cannot be related to the resulting gravitational-wave signal by a constant retarded time. This is due to the nontrivial dynamical spacetime curvature between the source and the signal. In a numerical-relativity simulation there is also some ambiguity in the black-hole dynamics, which depend on the gauge (coordinate) choices used in the numerical solution of Einstein's equations. It has been shown previously that a good approximation to the direction of the binary's time-dependent orbital angular momentum $\stackrel{^}{\mathbf{L}}(t)$ can be calculated from the gravitational-wave signal. This is done by calculating the direction that maximizes the quadrupolar ($\ensuremath{\ell}=2$, $|m|=2$) emission. The direction depends on whether we use the Weyl scalar ${\ensuremath{\psi}}_{4}$ or the gravitational-wave strain $h$, but these directions are nonetheless invariant for a given binary configuration. We treat the ${\ensuremath{\psi}}_{4}$-based direction as a proxy to $\stackrel{^}{\mathbf{L}}(t)$. We investigate how well the binary's orbital phase, ${\ensuremath{\phi}}_{\mathrm{orb}}(t)$, can also be estimated from the signal. For this purpose we define a quantity $\mathrm{\ensuremath{\Phi}}(t)$ that agrees well with ${\ensuremath{\phi}}_{\mathrm{orb}}(t)$. One application is to studies that involve injections of numerical-relativity waveforms into gravitational-wave detector data.