Engaging washers such as reaction washers and backup washers can include novel advantageous features. For example, an engaging washer can include an inner row of teeth and an outer row of teeth, where the teeth in the inner row can slope in a first circumferential direction to primarily engage a flange during bolt tightening, and the teeth in the outer row can slope in a second circumferential direction to primarily engage during bolt loosening. The engaging washers may also include an arched design to allow gradual increasing of contact surface areas as a clamping force in the washer gradually increases. Additionally, a reaction washer can include stepped castles for engaging a reaction socket. The castles may also include sloped or curved engaging surfaces that can provide decreased contact surface area and may also inhibit outward biasing of reaction fingers that are engaging the castles.