The SO(3) slide rule: a Wulff net with a turning overlay. Turn the overlay to rotate about your line of sight, and slide points along the small circles to rotate about the north–south axis. Those two moves build every rotation.
The chart. You look down on a sphere from above. A point (x, y, z) of the upper hemisphere is drawn at R·(x, y)/(1 + z): stereographic projection, which keeps angles and sends circles to circles. The rim is the horizon. With lower hemisphere turned on, points below are drawn as open circles at their antipodes.
The net. Meridians are great circles through N and S, spaced 10° apart by rotation about the N–S axis. Small circles are the paths points follow under that rotation. Along a meridian, small circles measure angle; along a small circle, meridians measure the rotation.
Two moves. Turn spins the overlay about the center, a rotation about the viewing axis z. Slide moves every point along its small circle, a rotation about the N–S axis y. A positive slide carries the center toward the east. Every rotation is turn · slide · turn: Euler angles.
Why two moves are enough. The Lie brackets of the two generators give the third direction: [L_z, L_y] = −L_x. That is the same reason parallel parking works.
The readout. The panel shows the total rotation as a unit quaternion (defined up to sign), as an axis and angle, as a matrix, and as z–y–z Euler angles. The Lab Manual has the paper version of this device and its exercises.