  We show that the Higgs boson structure functions observable in the inclusive
process $e^-e^+ \to H + {\rm "anything"}$ may reveal the presence of anomalous
contributions corresponding to several types of new physics partners, Higgs
boson compositeness or invisible (dark) matter. This could be done without
making a difficult or even an impossible experimental analysis of the contents
of the "anything". We give illustrations showing how the shapes of the various
structure functions containing such typical new contributions may differ from
the standard prediction and allow their identification.
