Asturias, the region in the north of Spain, is cider country. Good Asturian cider is very slightly effervescent, and most of the bubbles come from the way it's poured. The bottle is held high, the glass low. The cider comes down in a thin stream. See the above picture.
This is to say, true Asturian cider is not supposed to bubble like beer, much less cava. Nor is it supposed to be sweet. It's usually fairly dry. The experience of drinking a good sidra asturiana is much like drinking ribeiro, the slightly fizzy white wine from neighboring Galicia. Asturian cider is in fact an alternative to a simple country wine, and the perfect accompaniment to Asturian foods.
Jeannie, a reader of Simple Spanish Food, recommended to me a great cider bar, sidrería, called El Ñeru. While Ana and I were in Madrid in May, we swung by and had cider and toast smeared with cabrales cheese--the blue cheese from Asturias, one of the best possible tapas to go with Asturian cider. Thanks for the tip, Jeannie! El Ñeru is at Bordadores 5.
Another good sidrería in Madrid is Casa Mingo, at Paseo de la Florida 34, not far from the chapel San Antonio de La Florida. See Goya's frescoes of Madrileña angels, and then drink some cider.
The on-line Spanish food store La Tienda sells natural Asturian cider and cider glasses to serve it in. These glasses are not only authentic, they make it easier to pour the cider in a high arc without spilling too much. (Even pro cider pourers in Spain always seem to spill a little).
An Asturian-themed tapas party might include sliced chorizo quick-fried in olive oil, cabrales cheese, hazelnuts packed in Asturian honey, plenty of good bread (see my recipes for easy, no-knead Spanish-style breads), and Asturian cider.
A word of warning: you may come across the inexpensive, sweet, artificially carbonated bottles of Spanish cider. This is a completely different drink. This cider also comes from Asturias, but it is not traditional Asturian cider. It is sometimes used as a dessert wine, like a sweet cava. To be honest, I don't like it. This type of cider has a plastic champagne cork wired to the top of the bottle.