  Cosmic Microwave Background (CMB) polarimeters aspire to measure the faint
$B$-mode signature predicted to arise from inflationary gravitational waves.
They also have the potential to constrain cosmic birefringence which would
produce non-zero expectation values for the CMB's $TB$ and $EB$ spectra.
However, instrumental systematic effects can also cause these $TB$ and $EB$
correlations to be non-zero. In particular, an overall miscalibration of the
polarization orientation of the detectors produces $TB$ and $EB$ spectra which
are degenerate with isotropic cosmological birefringence, while also
introducing a small but predictable bias on the $BB$ spectrum. The \bicep
three-year spectra, which use our standard calibration of detector polarization
angles from a dielectric sheet, are consistent with a polarization rotation of
$\alpha = -2.77^\circ \pm 0.86^\circ \text{(statistical)} \pm 1.3^\circ
\text{(systematic)}$. We revise the estimate of systematic error on the
polarization rotation angle from the two-year analysis by comparing multiple
calibration methods. We investigate the polarization rotation for the \bicep
100 GHz and 150 GHz bands separately to investigate theoretical models that
produce frequency-dependent cosmic birefringence. We find no evidence in the
data supporting either these models or Faraday rotation of the CMB polarization
by the Milky Way galaxy's magnetic field. If we assume that there is no cosmic
birefringence, we can use the $TB$ and $EB$ spectra to calibrate detector
polarization orientations, thus reducing bias of the cosmological $B$-mode
spectrum from leaked $E$-modes due to possible polarization orientation
miscalibration. After applying this "self-calibration" process, we find that
the upper limit on the tensor-to-scalar ratio decreases slightly, from $r<0.70$
to $r<0.65$ at $95\%$ confidence.
