  Magnetic helicity, $H$, measures magnetic linkages in a volume. The early
theoretical development of helicity focused on magnetically closed systems in
$\mathcal{V}$ bounded by $\mathcal{S}$. For magnetically closed systems,
$\mathcal{V}\in\mathbb{R}^3=\mathcal{V}+\mathcal{V}^*$, no magnetic flux
threads the boundary, $\hat{\boldsymbol{n}}\cdot\boldsymbol{B}|_\mathcal{S}=0$.
Berger and Field (1984) and Finn and Antonsen (1985) extended the definition of
helicity to relative helicity, $\mathcal{H}$, for magnetically open systems
where magnetic flux may thread the boundary. Berger (1999,2003) expressed this
relative helicity as two gauge invariant terms that describe the self helicity
of magnetic field that closes inside $\mathcal{V}$ and the mutual helicity
between the magnetic field that threads the boundary $\mathcal{S}$ and the
magnetic field that closes inside $\mathcal{V}$. The total magnetic field that
permeates $\mathcal{V}$ entangles magnetic fields that are produced by current
sources $\boldsymbol{J}$ in $\mathcal{V}$ with magnetic fields that are
produced by current sources $\boldsymbol{J}^*$ in $\mathcal{V}^*$. Building on
this fact, we extend Berger's expressions for relative magnetic helicity to
eight gauge invariant quantities that simultaneously characterize both of these
self and mutual helicities and attribute their origins to currents
$\boldsymbol{J}$ in $\mathcal{V}$ and/or $\boldsymbol{J}^*$ in $\mathcal{V}^*$,
thereby disentangling the domain of origin for these entangled linkages. We
arrange these eight terms into novel expressions for internal and external
helicity (self) and internal-external helicity (mutual) based on their domain
of origin. The implications of these linkages for interpreting magnetic energy
is discussed and new boundary observables are proposed for tracking the
evolution of the field that threads the boundary.
