Our understanding of dust in the diffuse interstellar medium remains incomplete with regard to the structure, composition, size distribution, and alignment properties of its grains. Joint observations of reddening, starlight polarisation spectra, and polarised dust emission for individual sightlines provide essential constraints on such properties. We study a far-UV selected sample of 96 reddening curves, for which optical linear polarisation spectra were obtained with FORS at the VLT as part of the Large Interstellar Polarisation Survey (LIPS). Starlight polarisation spectra for 60 stars are presented in this work. These data are combined with Gaia distance estimates and Planck thermal dust emission. A three-component dust model is made publicly available. It consists of nanoparticles, amorphous grains, and micrometre-sized dust agglomerates, varying axial ratios, porosities, sizes, element abundances, and alignment efficiencies to match the observations. The diversity of reddening and polarisation spectra is well reproduced by prolate grains with typical axial ratios of two, a porosity of 10%, and high alignment efficiencies. Such efficiencies can be achieved with radiative torque alignment theory (RAT), but not with imperfect Davis-Greenstein (IDG) alignment, except when adjusting the magnetic-field orientation to maximise the polarisation. Micrometre-sized dust contributes wavelength-independent grey extinction in the optical, accounts for about one-third of the visual extinction, and carries one-third of the dust mass. A follow-up submillimetre survey with high-resolution polarimetry will further constrain grain shapes and alignment physics.