Abstract
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.