A compound miter combines two saw settings in a single cut: the miter table swings to an angle in plan while the blade tilts to a bevel angle, so the cut plane is skewed in both directions at once. It is what cutting crown molding flat on the saw table requires, because a piece that will lean off the wall at its spring angle meets the corner along a plane no single-angle cut can produce. The settings come from the compound-angle identities — miter = atan(sin S ÷ tan(C/2)), bevel = asin(cos S × cos(C/2)) — with S the spring angle and C the included corner angle; for standard 38°-spring crown in a square corner they evaluate to 31.62° and 33.86°, the crown detents machined into compound miter saws.
The angles are only half of a usable answer — the other half is orientation: inside and outside corners use identical miter and bevel values, and what changes is which way the table swings and which end of the cut is saved for the left- and right-hand pieces (cut face up, flat, top of the molding against the fence). The escape hatch from compound cutting entirely is the nested method: crown held upside-down and backwards against the fence takes a plain miter of (180° − C)/2 — a flat 45° for a square corner — and no bevel at all. Either way, saw detents, spring-angle tolerance in the stock, and out-of-square rooms each shift the fit, so the last half degree is always found on scrap.