The mechanical behaviour of ionically cross-linked alginate gels is investigated here in detail. To determine the range of linear response of the materials, uniaxial, unconfined compression and torsional deformation experiments are performed, obtaining both the stress-strain and the viscoelastic behaviour of the gels. On-line measurements of the radii of the cylindrical gel samples in these experiments are also reported. The linearity range in the gel mechanical response is found to be rather limited, up to 6% strain, at most, contrary to more optimistic conclusions usually reported in the literature. We confirm the presence of a stress-diffusion coupling phenomenon in our alginates, i.e., the migration of water from/into the gels in response to the applied deformation. A phenomenon of inner (constitutive) relaxation of the network component of the gels is also clearly identified, and observed to occur, in parallel with solvent diffusion, upon compression. At sufficiently longer times after a deformation step, syneresis is always observed, with concomitant nonstandard viscoelastic effects, such as the growth of a normal force in torsion, and a size dependent decay of the longitudinal force in compression. We applied a two-fluid model, recently developed by two of the present authors [D. Larobina and F. Greco, J. Chem. Phys., 2012, 136(13), 134904], to simulate the relaxation tests upon torsional and compressive deformations, and to fit our own experiments. The model is found to well describe the coupling between constitutive relaxation and diffusion, and to reproduce the available force and radii data before the advent of syneresis.