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Four getaways with a night away and three day trips from Madrid for Maundy Thursday and Good Friday: processions of international interest less than two hours away, Chinchón's living Passion and a national park for those who prefer birds.
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Cuenca's Holy Week is of International Tourist Interest, and its most singular procession, the Turbas, leaves El Salvador in the early hours of Good Friday amid drums and bugles. On those days the city also holds the Religious Music Week: book the hotel in the old town months ahead.
On Holy Saturday, as dusk falls, more than 250 locals act out the Passion around the Plaza Mayor and the streets of Chinchón, with free entry: it is a Festival of National Tourist Interest. Sleep in Aranjuez and arrive in good time, as the square fills up hours beforehand.

On Good Friday the Procession of the Pasos leaves the Plaza Mayor and ends in the Plaza de la Artillería, at the foot of the Aqueduct; Segovia's Holy Week is of National Tourist Interest. The balloon, if the wind allows, flies at dawn with the city still silent.

For those who prefer birds to floats: in spring black vultures, imperial eagles and black storks nest on the park's cliffs, and from March to June there are free guided walks. The rural house in Villarreal de San Carlos and the 4x4 are booked before anything else.

Salamanca has a Holy Week of International Tourist Interest, with more than twenty processions through the old town. On Maundy Thursday and Good Friday the cathedrals hold services, so the visit goes in the morning; stay for dinner, as the floats go out at night and some at midnight.

In the early hours of Good Friday, the Vía Crucis de Penitencia circles the wall of Ávila with the Cristo de los Ajusticiados; Ávila's Holy Week is of International Tourist Interest. By day, the wall walk and San Vicente; at dusk, the Cuatro Postes with the wall lit up.

Toledo's Holy Week is of International Tourist Interest and its processions leave from the cathedral or pass beside it, so the services change the visiting hours. Buy the cathedral with a timed ticket, check the traffic closures in the old town and park below, by the river.
Questions
No. In the Community of Madrid, Maundy Thursday and Good Friday are holidays; Easter Monday is a holiday in other communities, such as Catalonia, the Valencian Community, Navarre and the Basque Country.
The Turbas of Cuenca and the Vía Crucis around the wall of Ávila, both in the early hours of Good Friday. From Madrid you cannot arrive in time in the morning.
On Holy Saturday at dusk, in the Plaza Mayor and the streets of the centre. It is free and more than 250 locals take part.
Yes, to Cuenca, Segovia, Toledo and Salamanca by fast train and to Aranjuez by Cercanías. There is a bus to Chinchón, and for Monfragüe you need a car.
Geocaliahttps://www.geocalia.com/en/spain/community-of-madrid/madrid/getaways/easter-holy-week