In contrast to conventional RIS, the scattering matrix of a non-reciprocal RIS (NR-RIS) is non-symmetric, leading to differences in the uplink and the downlink components of NR-RIS cascaded channels. In this paper, a physically-consistent device model is proposed in which an NR-RIS is composed of multiple groups of two-port elements inter-connected by non-reciprocal devices. The resulting non-reciprocal scattering matrix is derived for various cases including two-element groups connected with isolators or gyrators, and general three-element groups connected via circulators. Signal models are given for NR-RIS operating in either reflecting-only or simultaneously transmitting and reflecting modes. The problem of NR-RIS design for non-reciprocal beamsteering is formulated for three-element circulator implementations, and numerical results confirm that non-reciprocal beamsteering can be achieved with minimal sidelobe power. We also show that our physically consistent NR-RIS architecture is effective in implementing channel reciprocity attacks, achieving similar performance to that with idealized NR-RIS models.