The Ópera quarter is monumental Madrid at its most deliberate — the Bourbons' answer to the Habsburgs' tangle of lanes, all axes and stone and sightlines. Two and a half kilometres from the opera house to the river, downhill at the end, best in the morning before the palace queue builds and the Sabatini gardens fill.
Start at Plaza de Isabel II in front of the Teatro Real, then walk west into Plaza de Oriente, the crescent of formal hedges and over-heavy stone kings. Ahead is the Palacio Real, the largest royal palace in Western Europe, raised after the old fortress burned in 1734. The Plaza de la Armería courtyard is free; the state rooms are not — take the scale from the courtyard and decide. Across it stands the Catedral de la Almudena, begun in 1879 and finished in 1993, with a painted modern ceiling that catches everyone off guard.
Round the north side into the Jardines de Sabatini — clipped hedges, a still pool, the best free bench in the city — then drop down Cuesta de San Vicente to Príncipe Pío and the line home.