  The search for di-Higgs production at the LHC in order to set limits on Higgs
trilinear coupling and constraints on new physics is one of the main
motivations for the LHC high luminosity phase. Recent experimental analyses
suggest that such analyses will only be successful if information from a range
of channels is included. We therefore investigate di-Higgs production in
association with two hadronic jets and give a detailed discussion of both the
gluon- and weak boson fusion contributions, with a particular emphasis on the
phenomenology with modified Higgs trilinear and quartic gauge couplings. We
perform a detailed investigation of the full hadronic final state and find that
$hhjj$ production should add sensitivity to a di-Higgs search combination at
the HL-LHC with 3 ab$^{-1}$. Since the WBF and GF contributions are sensitive
to different sources of physics beyond the Standard Model, we devise search
strategies to disentangle and isolate these production modes. While gluon
fusion remains non-negligible in WBF-type selections, sizeable new physics
contributions to the latter can still be constrained. As an example of the
latter point we investigate the sensitivity that can be obtained for a
measurement of the quartic Higgs-gauge boson couplings.
