  We present interferometeric measurements of the $f=1$ to $f=2$
inter-hyperfine scattering lengths in a single-domain spinor Bose-Einstein
condensate of $^{87}$Rb. The inter-hyperfine interaction leads to a strong and
state-dependent modification of the spin-mixing dynamics with respect to a
non-interacting description. We employ hyperfine-specific Faraday-rotation
probing to reveal the evolution of the transverse magnetization in each
hyperfine manifold for different state preparations, and a comagnetometer
strategy to cancel laboratory magnetic noise. The method allows precise
determination of inter-hyperfine scattering length differences, calibrated to
intra-hyperfine scattering length differences. We report
$(a_{3}^{(12)}-a_{2}^{(12)})/(a_{2}^{(1)}-a_{0}^{(1)})=-1.27(15)$ and
$(a_{1}^{(12)}-a_{2}^{(12)})/(a_{2}^{(1)}-a_{0}^{(1)})=-1.31(13)$, limited by
atom number uncertainty. With achievable control of atom number, we estimate
precisions of $ \approx 0.3\%$ should be possible with this technique.
