Beyond diagonal reconfigurable intelligent surfaces (BD-RIS) is a new advance in RIS techniques that introduces reconfigurable inter-element connections to generate scattering matrices not limited to being diagonal. BD-RIS has been recently proposed and proven to have benefits in enhancing channel gain and enlarging coverage in wireless communications. Uniquely, BD-RIS enables reciprocal and non-reciprocal architectures characterized by symmetric and non-symmetric scattering matrices. However, the performance benefits and new use cases enabled by non-reciprocal BD-RIS for wireless systems remain unexplored. This work takes a first step toward closing this knowledge gap and studies the non-reciprocal BD-RIS in full-duplex systems and its performance benefits over reciprocal counterparts. We start by deriving a general RIS aided full-duplex system model using a multiport circuit theory, followed by a simplified channel model based on physically consistent assumptions. With the considered channel model, we investigate the effect of BD-RIS non-reciprocity and identify the theoretical conditions for reciprocal and non-reciprocal BD-RISs to simultaneously achieve the maximum received power of the signal of interest in the uplink and the downlink. Simulation results validate the theories and highlight the significant benefits offered by non-reciprocal BD-RIS in full-duplex systems. The significant gains are achieved because of the non-reciprocity principle which implies that if a wave hits the non-reciprocal BD-RIS from one direction, the surface behaves differently than if it hits from the opposite direction. This enables an uplink user and a downlink user at different locations to optimally communicate with the same full-duplex base station via a non-reciprocal BD-RIS, which would not be possible with reciprocal surfaces.