  We present evidence for scale-independent misalignment of interstellar dust
filaments and magnetic fields. We estimate the misalignment by comparing
millimeter-wave dust-polarization measurements from Planck with filamentary
structures identified in neutral-hydrogen (HI) measurements from HI4PI. We find
that the misalignment angle displays a scale independence (harmonic coherence)
for features larger than the HI4PI beam width ($16.2'$). We additionally find a
spatial coherence on angular scales of $\mathcal{O}(1^\circ)$. We present
several misalignment estimators formed from the auto- and cross-spectra of
dust-polarization and HI-based maps, and we also introduce a map-space
estimator. Applied to large regions of the high-Galactic-latitude sky, we find
a global misalignment angle of $\sim 2^\circ$, which is robust to a variety of
masking choices. By dividing the sky into small regions, we show that the
misalignment angle correlates with the parity-violating $TB$ cross-spectrum
measured in the Planck dust maps. The misalignment paradigm also predicts a
dust $EB$ signal, which is of relevance in the search for cosmic birefringence
but as yet undetected; the measurements of $EB$ are noisier than of $TB$, and
our correlations of $EB$ with misalignment angle are found to be weaker and
less robust to masking choices. We also introduce an HI-based dust-polarization
template constructed from the Hessian matrix of the HI intensity, which is
found to correlate more strongly than previous templates with Planck dust $B$
modes.
