The Dirac belt trick produces a nice 3-dimensional geometric object with symmetry group $Spin(3) = SU(2)$: a 2-sphere with a properly embedded framed ray (usually presented by using orientations to reduce the framing of the ray to a single normal vector field, then integrating this to give a "belt" of finite width), with the ray or "belt" regarded up to smooth isotopies fixing the sphere and trivial at infinity.
A naive generalization to higher dimensions using a (2-dimensional) "belt" does not work to produce an object with symmetry group $Spin(n)$ since the "belt" can always be untwisted.
My question:
Is there any properly embedded subspace of ${\mathbb R}^n$ with boundary on $S^{n-1}$, possibly equipped with an auxiliary structure like a framing of its normal bundle, such that when the subspace is regarded up to isotopies fixing $S^{n-1}$ and trivial at infinity, the symmetry group of the "sphere and subspace" is $Spin(n)$?
I am particularly interested in the case of $n=4$.