Madridman M.
Yelp
Formerly known (before 2011) as the "Palacio de Comunicaciones", which was the old post office since 1919, the now-named "Palacio de Cibeles" is now Madrid's main City Hall, transferring most of the offices - including that of the Mayor of Madrid - from its former location on the Plaza de la Villa.
The Palacio de Cibeles is located on the plaza by the same name, the "Plaza de Cibeles", centered by a fountain and lion-towed chariot carrying the goddess Cibeles, at the 4-way intersection of the Calle Alcalá (east-west), Paseo del Prado, (south) and the Paseo de Recoletos (north) - which becomes the Paseo de la Castellana further north.
The ground floor of the main building is large public space with two reading spaces with sofas and chairs, newspapers and magazines, & free wi-fi internet, iPads under glass for public use (10-20% in working order), (clean-and-modern) PUBLIC RESTROOMS, and audio/video exhibits of the building's construction and history. The central, open space reaches up 4 floors in the main building, with beautiful stained-glass on several ceilings. The public is free to go up, essentially, to the 6th floor. The first 4 floors have several other exhibits about the building.
One can visit the center clock tower's observation deck on level 8, but first must pass through metal detectors just to the right of the main, street-level entrance, then acquiring a FREE pass at the main-floor desk (level 1). Self-guided tours take place every 15-minutes and your ticket will assign the next available tour entry time. With the ticket, take the elevator up to floor 6E and wait in line for your time's tour. Guards will call out the tour entry (in Spanish) and the public can ascend by stairs (faster) or (wait for the) elevator to the outdoor, 8th floor observation deck.
From the observation deck, one has a wonderful 360º video of Madrid, overlooking the busy Plaza de Cibeles below, the tower of the Círculo de Bellas Artes on Calle de Alcalá to the west, and up the Paseo de Recoletos/Castellana to the north while circling the clock tower. At near the end of your 15-minutes, the guard will call out for the deck to be vacated in order to allow the next group's entry. Generally speaking, access is closed from 2pm-4pm. See (my) photos of the inside and out.
This is the BEST FREE observation deck in Madrid, folks!
Saludos, MadridMan @ you-know-where!