Palacio de Cibeles, a.k.a The Palace of Communications or simply Madrid's City Hall is one of the most majestic architectural landmarks of the Spanish capital.
The entrance for the first six floors of the building is free, so don't hesitate to enter and take a look. The large cultural space inside is called CentroCentro (http://www.centrocentro.org/), and it will probably intrigue you too.
Anyway, if you care to enjoy the superb view from the rooftop terrace & bar, you'll have to pay the €6 entrance fee.