  Gedanken experiments in quantum gravity motivate generalised uncertainty
relations (GURs) implying deviations from the standard quantum statistics close
to the Planck scale. These deviations have been extensively investigated for
the non-spin part of the wave function but existing models tacitly assume that
spin states remain unaffected by the quantisation of the background in which
the quantum matter propagates. Here, we explore a new model of nonlocal
geometry in which the Planck-scale smearing of classical points generates GURs
for angular momentum. These, in turn, imply an analogous generalisation of the
spin uncertainty relations. The new relations correspond to a novel
representation of {\rm SU(2)} that acts nontrivially on both subspaces of the
composite state describing matter-geometry interactions. For single particles
each spin matrix has four independent eigenvectors, corresponding to two
$2$-fold degenerate eigenvalues $\pm (\hbar + \beta)/2$, where $\beta$ is a
small correction to the effective Planck's constant. These represent the spin
states of a quantum particle immersed in a quantum background geometry and the
correction by $\beta$ emerges as a direct result of the interaction terms. In
addition to the canonical qubits states, $\ket{0} = \ket{\uparrow}$ and
$\ket{1} = \ket{\downarrow}$, there exist two new eigenstates in which the spin
of the particle becomes entangled with the spin sector of the fluctuating
spacetime. We explore ways to empirically distinguish the resulting `geometric'
qubits, $\ket{0'}$ and $\ket{1'}$, from their canonical counterparts.
