Maps and compass go together like ham and eggs. Actually, if I had to choose between the two, I would rather have a map, at least for the country in which I do most of my rambling, the SW United States. In most areas here, it is very easy to orient yourself by visual references without recourse to a compass. In general, a USGS topo map, 7.5 minute series is the best for foot travel. But about every five years or so, descending fog or something restricts visibility and renders a compass an absolute necessity.
There are many ways in which a compass needle can be influenced to give an erroneous reading - locally carried ferrous objects (like the screws in your eyeglasses!) or local bodies of ore. Specifically, if you are climbing Orizaba, your compass will be out about 70 degrees. I had a long discussioon with my climbing partner who was using his compass, that was only settled when I pointed out the location of the North Star. But yes, basically the compass needle is usually right, and definitely right far more often than ones instincts.
I currently carry a Suunto MC-2 which has an inclinometer and mirror. The mirror definitely works well for signalling and even for shaving. Accurate, compact, and light.