In this paper, we consider a cooperative underlay cognitive radio network in which the primary network (PN) consists of a transmitter and receiver and the secondary network (SN) has K bidirectional half-duplex relays. In the SN, two secondary transceivers adopt multiple access broadcast protocol for the secondary data transmission and at each bidirectional relay, there exist two buffers of size L data elements. Hence, each relay can store the incoming secondary data and retransmit it in an appropriate time slot later. We propose a novel buffer-aided bidirectional relay selection policy with secondary power minimization and successive interference cancellation in which the interference between the PN and SN is eliminated. Since buffers are used at the relays, data transmission in the SN is not limited to a predefined schedule. Hence, at each time slot, based on the instantaneous buffer state information of the relays and the instantaneous or statistical channel state information of the involved links, the SN makes a decision. The SN decides optimally when to use one of the relays for the multiple access, use one of the relays for the broadcast mode or be silent provided that the data transmission in both the PN and SN are error free and the secondary power expenditure is minimized. Simulation results show that the proposed scheme minimizes the secondary power expenditure, and achieves up to 40% improvement in the secondary throughput for 6 middle relays compared to the other recently proposed policies without buffer.