A bidirectional integrated sensing and communication (ISAC) system is proposed, in which a pair of transceivers carry out two-way communication and mutual sensing. Both full-duplex and half-duplex operations in narrowband and wideband systems are conceived for the bidirectional ISAC. 1) For the narrowband system, the conventional full-duplex and half-duplex operations are redesigned to take into account sensing echo signals. Then, the transmit beamforming design of both transceivers is proposed for addressing the sensing and communication (S&C) tradeoff. A one-layer iterative algorithm relying on successive convex approximation (SCA) is proposed to obtain Karush-Kuhn-Tucker (KKT) optimal solutions. 2) For the wideband system, the new full-duplex and half-duplex operations are proposed for the bidirectional ISAC. In particular, the frequency-selective fading channel is tackled by delay pre-compensation and path-based beamforming. By redesigning the proposed SCA-based algorithm, the KKT optimal solutions for path-based beamforming for characterizing the S&C tradeoff are obtained. Finally, the numerical results show that: i) For both bandwidth scenarios, the existence of the interference introduced by sensing results in full-duplex may not always outperform half-duplex, especially in the sensing-prior regime or when the communication channel is line-of-sight-dominated; and ii) For both duplex operations, it is sufficient to reuse communication signals for sensing in the narrowband system, while an additional dedicated sensing signal is required in the wideband system.