  We develop a new approach to the study of spectral asymmetry. Working with
the operator $\operatorname{curl}:=*\mathrm{d}$ on a connected oriented closed
Riemannian 3-manifold, we construct, by means of microlocal analysis, the
asymmetry operator -- a scalar pseudodifferential operator of order $-3$. The
latter is completely determined by the Riemannian manifold and its orientation,
and encodes information about spectral asymmetry. The asymmetry operator
generalises and contains the classical eta invariant traditionally associated
with the asymmetry of the spectrum, which can be recovered by computing its
regularised operator trace. Remarkably, the whole construction is direct and
explicit, and does not involve analytic continuation or algebraic topology.
