In this paper, we consider a two-transmitter-two-receiver cognitive interference channel with causal bidirectional cooperation (CIFC-CBC), where the transmitters operate in a full-duplex mode and cooperate with each other in a causal way. Cooperation between the transmitters is realized using a strategy that combines both decode-and-forward (DF) and compress-and-forward (CF) schemes. We derive an achievable rate region for the considered system based on a proposed coding scheme that jointly implements rate splitting, block Markov coding, superposition coding, Gel'fand-Pinsker coding, and backward decoding. We then apply the derived rate region result to Gaussian channels and show via numerical results that the derived rate region extends the rate regions achieved by pure DF- and pure CF-based cooperation.