Madrid's cathedral is Santa María la Real de La Almudena. Our guidebook pans it as a tourist site, but I found it fascinating and, in parts, moving.
One reason the guidebook pans it, is that the cathedral is new. Though it was started in the 1800s, it wasn't completed until 1993, when it was finally consecrated.
The cathedral is named for The Virgin of Alumdena -- an very old icon. Legend has it that it was carved by St. Nicodemus (who assisted at Jesus's burial) and was painted by St. Luke. Curiously, the name -- Alumdena -- comes from the Arabic "the fortress." The icon is on display at the cathedral.
What I liked about the cathedral was the sometimes jarring modern/pop art decorations. The photos show some of these.