This paper investigates the performance of reconfigurable intelligent surface assisted two-way non-orthogonal multiple access (RIS-TW-NOMA) networks, where a pair of users exchange their information through a RIS. The influence of imperfect successive interference cancellation on RIS-TW-NOMA is taken into account. To evaluate the potential performance of RIS-TW-NOMA, we derive the exact and asymptotic expressions of outage probability and ergodic rate for a pair of users. Based on the analytical results, the diversity orders and high signal-to-noise ratio (SNR) slopes are obtained in the high SNR regime, which are closely related to the number of RIS elements. Additionally, we analyze the system throughput and energy efficiency of RIS-TW-NOMA networks in both delay-limited and delay-tolerant transmission modes. Numerical results indicate that: 1) The outage behaviors and ergodic rate of RIS-TW-NOMA are superior to that of RIS-TW-OMA and two-way relay OMA (TWR-OMA); 2) As the number of RIS elements increases, the RIS-TW-NOMA networks are capable of achieving the enhanced outage performance; and 3) By comparing with RIS-TW-OMA and TWR-OMA networks, the energy efficiency and system throughput of RIS-TW-NOMA has obvious advantages.