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The stiffness modulation mechanism for soft robotics has gained considerable attention to improve deformability, controllability, and stability. However, for the existing stiffness soft actuator, high lateral stiffness and a wide range of bending stiffness are hard to be provided at the same time. This paper presents a bioinspired bidirectional stiffening soft actuator (BISA) combining the air-tendon hybrid actuation (ATA) and a bone-like structure (BLS). The ATA is the main actuation of the BISA, and the bending stiffness can be modulated with a maximum stiffness of about 0.7 N/mm and a maximum magnification of 3 times when the bending angle is 45 deg. Inspired by the morphological structure of the phalanx, the lateral stiffness can be modulated by changing the pulling force of the BLS. The lateral stiffness can be modulated by changing the pulling force to it. The actuator with BLSs can improve the lateral stiffness about 3.9 times compared to the one without BLSs. The maximum lateral stiffness can reach 0.46 N/mm. And the lateral stiffness can be modulated decoupling about 1.3 times (e.g., from 0.35 N/mm to 0.46 when the bending angle is 45 deg). The test results show the influence of the rigid structures on bending is small with about 1.5 mm maximum position errors of the distal point of actuator bending in different pulling forces. The advantages brought by the proposed method enable a soft four-finger gripper to operate in three modes: normal grasping, inverse grasping, and horizontal lifting. The performance of this gripper is further characterized and versatile grasping on various objects is conducted, proving the robust performance and potential application of the proposed design method.