Large changes in the rheology of high-internal phase aqueous-in-oil emulsions (HIPEs) using an oil-soluble polyisobutylene-based primary surfactant (PIBSA) are provoked by very small quantities of water-soluble polyamide-based cosurfactants (PAM with C(12), C(14), and C(16) tails). The structural origin of this was studied using small-angle neutron scattering (SANS) from sheared emulsions, with simultaneous in situ rheology measurements. The PAM drastically lowers the droplet oil interfacial tension by displacing PIBSA, causing large droplet deformation under shear and much lowered emulsion yield stress. With PAM, the surfactant monolayer at the droplet surface becomes more responsive to droplet shape change and redistributes in response to shear which the PIBSA-only system does not. Although it is oil-insoluble, PAM also reaches the nanoscale PIBSA micelles in the oil phase, changing micelle size and content in ways predictable from the hydrophilicity of the different PAMs. PAM does not, however, strongly affect the viscosities at high shear rates; shear thinning and thickening are unaffected. Droplet size, droplet-droplet flattening, and linkage determine the viscosities observed, more so than, droplet-oil interfacial tension. We infer from this that the droplet motion under shear does not involve much transient droplet deformation as the droplets move by each other.