  We reinterpret jet clustering as an axis-finding procedure which, along with
the proton beam, defines the virtual-photon transverse momentum $q_T$ in deep
inelastic scattering (DIS). In this way, we are able to probe the nucleon
intrinsic structure using jet axes in a fully inclusive manner, similar to the
Drell-Yan process. We present the complete list of azimuthal asymmetries and
the associated factorization formulae at leading power for deep-inelastic
scattering of a nucleon. The factorization formulae involve both the
conventional time-reversal-even (T-even) jet function and the T-odd one, which
have access to all transverse-momentum-dependent parton distribution functions
(TMD PDFs) at leading twist. Since the factorization holds as long as $q_T \ll
Q$, where $Q$ is the photon virtuality, the jet-axis probe into the nucleon
structure should be feasible for machines with relatively low energies such as
the Electron-Ion Collider in China (EicC). We show that, within the
winner-take-all (WTA) axis-finding scheme, the coupling between the T-odd jet
function and the quark transversity or the Boer-Mulders function could induce
sizable azimuthal asymmetries at the EicC, the EIC and HERA. We also give
predictions for the azimuthal asymmetry of back-to-back dijet production in
$e^+e^-$ annihilation at Belle and other energies.
