  The CMB $B$-mode polarisation signal -- both the primordial gravitational
wave signature and the signal sourced by lensing -- is subject to many
contaminants from systematic effects. Of particular concern are systematics
that result in mixing of signals of different ``spin'', particularly leakage
from the much larger spin-0 intensity signal to the spin-2 polarisation signal.
We present a general formalism, which can be applied to arbitrary focal plane
setups, that characterises signals in terms of their spin. We provide general
expressions to describe how spin-coupled signals observed by the detectors
manifest at map-level, in the harmonic domain, and in the power spectra,
focusing on the polarisation spectra -- the signals of interest for upcoming
CMB surveys. We demonstrate the presence of a previously unidentified
cross-term between the systematic and the intrinsic sky signal in the power
spectrum, which in some cases can be the dominant source of contamination. The
formalism is not restricted to intensity to polarisation leakage but provides a
complete elucidation of all leakage including polarisation mixing, and applies
to both full and partial (masked) sky surveys, thus covering space-based,
balloon-borne, and ground-based experiments. Using a pair-differenced setup, we
demonstrate the formalism by using it to completely characterise the effects of
differential gain and pointing systematics, incorporating both intensity
leakage and polarisation mixing. We validate our results with full time ordered
data simulations. Finally, we show in an Appendix that an extension of simple
binning map-making to include additional spin information is capable of
removing spin-coupled systematics during the map-making process.
