Structural seismic interpretation and quantitative characterization are historically intertwined processes. The latter provides estimates of properties of the subsurface which can be used to aid structural interpretation alongside the original seismic data and a number of other seismic attributes. In this work, we redefine this process as a inverse problem which tries to jointly estimate subsurface properties (i.e., acoustic impedance) and a piece-wise segmented representation of the subsurface based on user-defined macro-classes. By inverting for the quantities simultaneously, the inversion is primed with prior knowledge about the regions of interest, whilst at the same time it constrains this belief with the actual seismic measurements. As the proposed functional is separable in the two quantities, these are optimized in an alternating fashion, where each subproblem is solved using a Primal-Dual algorithm. Subsequently, each class is used as input to a workflow which aims to extract the perimeter of the detected shapes and to produce unique horizons. The effectiveness of the proposed method is illustrated through numerical examples on synthetic and field datasets.