A lightning discharge in a thundercloud usually starts with a locally breakdown process (preliminary breakdown) followed by a widely extending leader process. In the early 1960 s, from the view of fundamental electrostatics[] suggested that the lightning leader is initiated by an electrodeless discharge with a zero-net-charge conductive channel extending bidirectionally in the ambient electric field of the thundercloud, i.e.the bidirectional leader theory. However, the bidirectional leader theory has just been recognized by lightning researchers since the late 1980 s, when airplane[] and rocket-triggered lightning experiments have proven that the bidirectional leader theory provides the best common physical basis for explaining a variety of lightning processes. Nevertheless, challenges still remain in other properties of the bidirectional leader theory, such as the polarity asymmetry, the sustainability, instability and restrike of a leader channel, which are all the key concerns of lightning researches. In the present paper, we first briefly review the concept and development of the bidirectional leader theory and its appearances in various lightning processes, especially in the negative stepped leader and recoil leader process. By reviewing a variety of field observation data of inception thresholds and propagation properties of long gap spark discharges, we put forward and emphasize an alternative polarity asymmetry of the lightning leader:the polarity asymmetry of continuity, i.e., the continuity of positive leader and the intermittency of negative leader. Based on the detailed discussion, we then argue that the stepwise positive leader may be restrikes due to temporary instability of a continuous positive leader, while the intermittency of negative leader manifests not only a relatively regular stepped leader but also a more erratic negative recoil leader. We further propose that the positive leader-streamer may play a unique role in igniting all kinds of intermittent events in lightning, including the initiation of lightning with the fast positive streamer, the formation of space leader in front of a negative stepped leader with the secondary positive streamer, and the reformation of recoil-leader in an instability leader channel due to its low threshold of inception and propagation. Finally, we discuss the self-adjustable ability of a bidirectional leader, suggesting that the potential drop along the streamer channel in front of the leader tip may be an important factor for stabilizing the bidirectional leader channel.