When modelling ice flows there are several aspects which are poorly constrained by observations, in particular parameters related to ice rheology and basal sliding. In computational ice flow models, inversion methods are frequently used to estimate the spatial distribution of these hidden fields. These methods use surface measurement data in combination with a forward model of the ice dynamics that relate the hidden fields to the surface fields. In this study we approximate the forward model using first-order linear perturbation theory to gain insights into our ability to extract information about the ice viscosity at the same time as basal slipperiness, and to understand the theoretical limitations. We frame the inversion problem in terms of a Gaussian maximum a-posteriori estimation with explicitly stated priors for the hidden fields. We illustrate the inversion behaviour with perturbations applied to flow down a laterally confined channel, where both viscosity and slipperiness can play a significant role in the ice sheet dynamics. Our results indicate that it is possible to extract information about the viscosity field at the same time as estimating the basal slipperiness, with strong horizontal gradients in the surface velocity field essential for good viscosity retrieval. We recommend always inverting for basal slipperiness and viscosity together over grounded ice areas in ice-sheet models, and explicitly recognising prior knowledge of the hidden fields in the retrieval through either the inclusion of appropriate priors and regularisation.