  We investigate some geometric properties of the $\operatorname{curl}$
operator, based on its diagonalizationand its expression as a non-local
symmetry of the pseudo-derivative $(-\Delta)^{1/2}$ among divergence-free
vector fieldswith finite energy. In this context, we introduce the notion of
spin-definite fields, i.e. eigenvectorsof
$(-\Delta)^{-1/2}\operatorname{curl}$.The two spin-definite components of a
general 3D incompressible flow untangle the right-handed motion from the
left-handed one. Having observed that the non-linearity of Navier-Stokes has
the structure of a cross-productand its weak (distributional) form is a
determinant that involves the vorticity, the velocity and a test function,we
revisit the conservation of energy and the balance of helicity in a geometrical
fashion. We show that in the caseof a finite-time blow-up, both spin-definite
components of the flow will explose simultaneously and with equal rates,i.e.
singularities in 3D are the result of a conflict of spin, which is impossible
in the poorer geometry of 2D flows.We investigate the role of the local and
non-local determinants $$\int_0^T\int_{\mathbb{R}^3}\det(\operatorname{curl} u,
u, (-\Delta)^{\theta} u)$$ and their spin-definite counterparts, which drive
the enstrophy and, more generally, are responsible forthe regularity of the
flow and the emergence of singularities or quasi-singularities.As such, they
are at the core of turbulence phenomena.
