I have to say that while making my way through Madrid, my two main tools for getting around were (unsurprisingly) a map and a compass. I’m using the Lonely Planet ‘Spain’ guidebook which has good maps for central Madrid (but not so good for areas outside of the immediate center). In general, the map has proven to be quite accurate with the exception of a mis-marked one way street which caused me no end of grief on my bike (gotta write the editors about that error).
However, the real gem has been the compass. I have a basic handheld compass (I think it’s a Polaris but I’m not sure) that cost me maybe 8 bucks. I found that when I was using the map alone, regardless of whether I’m biking or walking, that my trips took roughly twice as long. That’s mainly because I had to stick to main streets to follow the map. Once I remembered that I brought my compass, it was much easier to get around because I could follow any street and keep my direction. Plus, the compass has a nice sharp edge (for tracing on a map) that I felt comfortable holding while I inadvertently stumbled into a less well-lit area (admittedly stupid).
Obvious? Sure, but I’d never really used a compass for navigating a city before and didn’t think it would be that useful. I was certainly wrong!