A lot of exercises, I get the impression, require a flat back. Like if you're doing a front lever or back lever in a "V" shape with your hips sagging your back isn't straight. And I think you're supposed to have a straight back for planks, dead lifts, etc.
Anyway, if you don't have a mirror, what's a good way to figure out if your back is flat?
You can lay down flat on your stomach and then lift your feet up. You don't only lift your toes off the ground, but you also lift your feet up to the point where your knees are also leaving the ground. You then repeat this for several repetitions. This will cause you to feel a certain burning sensation in the lower back muscles. The idea is, once you feel these muscles working you will be more in tune with them and as a result know when they are flat. Mark Rippetoe uses this technique with people who have a hard time getting their back flat.
http://startingstrength.com/index.php/site/video/platform_lower_back_position_control
In this video you will see him demonstrate this technique amongst others. This video is on the longer side (11 minutes), but really worth the watch as it should answer your question.
The first step is to discover the sensation of a flat back, of firing your lower back muscles, of arching your back, and of keeping or losing your back arch while the hips move. Practice "supermans" on the floor, lifting your legs and chest off the ground. Get into what you think is proper position, then feel with your hand or look in a mirror.
Reinforce this sensation using Romanian deadlifts and by squatting while holding a dowel or broomstick up and down your back, touching your sacrum, between your shoulderblades, and the back of your head.
Once you've achieved sensation of proper back position, it's important to get occasional feedback on how your proprioception is doing. I'll ask a knowledgeable lifter to take a look at my deadlift (or clean or squat) and let me know how my back arch looks. I'll also set up a camera with a tripod and see how it looks to me.
