Robot arms that assist humans should be able to pick up, move, and release everyday objects. Today's assistive robot arms use rigid grippers to pinch items between fingers; while these rigid grippers are well suited for large and heavy objects, they often struggle to grasp small, numerous, or delicate items (such as foods). Soft grippers cover the opposite end of the spectrum; these grippers use adhesives or change shape to wrap around small and irregular items, but cannot exert the large forces needed to manipulate heavy objects. In this paper we introduce RIgid-SOft (RISO) grippers that combine switchable soft adhesives with standard rigid mechanisms to enable a diverse range of robotic grasping. We develop RISO grippers by leveraging a novel class of soft materials that change adhesion force in real-time through pneumatically controlled shape and rigidity tuning. By mounting these soft adhesives on the bottom of rigid fingers, we create a gripper that can interact with objects using either purely rigid grasps (pinching the object) or purely soft grasps (adhering to the object). This increased capability requires additional decision making, and we therefore formulate a shared control approach that partially automates the motion of the robot arm. In practice, this controller aligns the RISO gripper while inferring which object the human wants to grasp and how the human wants to grasp that item. Our user study demonstrates that RISO grippers can pick up, move, and release household items from existing datasets, and that the system performs grasps more successfully and efficiently when sharing control between the human and robot. See videos here: https://youtu.be/5uLUkBYcnwg