The iPhone 4S, iPhone 5, and iPhone 5S resting on the same flat surface. One of these things is not like the others.
Following numerous complaints about the issue on Apple's support forums, Gizmodo looked into reports that the motion sensors on the new iPhone 5S are not calibrated properly. By comparing the iPhone 5S to an iPhone 5 and also to several real-world tools that the phone replicates with motion sensors (including an actual compass and a spirit level), Gizmodo found that the gyroscope, compass, and accelerometer in two separate iPhone 5Ses do indeed appear to be off by a small but consistently measurable amount.
We confirmed Gizmodo's findings when we compared our own iPhone 5S to an iPhone 5 and an iPhone 4S, all running iOS 7.0.2—as you can see by the picture above, the iPhone 4S and iPhone 5 resting on our desk recognize that they're resting on a flat surface while the iPhone 5S does not. We re-calibrated the Compass app several times and consistently got the same results. The severity of the problem appears to vary from phone to phone—some posters in a MacRumors forum thread on the topic claim that their phones are off by four or six degrees (instead of two, as with our iPhone 5S and with Gizmodo's), while others report that they no longer had the issue after getting a replacement phone from Apple. The relatively wide range of inaccuracies being reported and the fact that the iPhone 5 and 4S aren't affected point to this being a hardware issue and not a software issue, though without a statement for Apple it's difficult to say for sure.
Beyond the inaccuracies in the compass app, the actual real-world impact of this issue is small. While Gizmodo's videos do show that the sensors can cause problems for games if you have the phone resting flat on a table, most people are going to subconsciously adjust the phone to compensate for the slight inaccuracies. In other games, the problems aren't large enough to register—the line in Ridiculous Fishing didn't drift in either direction if our 5S was standing upright on a flat surface. We've contacted Apple for comment on the issue and will update this post if we receive a response. In the meantime, try not to use your iPhone 5S to hang any curtains or anything.