We propose uncalibrated photometric stereo methods that address the problem due to unknown isotropic reflectance. At the core of our methods is the notion of "constrained half-vector symmetry" for general isotropic BRDFs. We show that such symmetry can be observed in various real-world materials, and it leads to new techniques for shape and light source estimation. Based on the 1D and 2D representations of the symmetry, we propose two methods for surface normal estimation; one focuses on accurate elevation angle recovery for surface normals when the light sources only cover the visible hemisphere, and the other for comprehensive surface normal optimization in the case that the light sources are also non-uniformly distributed. The proposed robust light source estimation method also plays an essential role to let our methods work in an uncalibrated manner with good accuracy. Quantitative evaluations are conducted with both synthetic and real-world scenes, which produce the state-of-the-art accuracy for all of the non-Lambertian materials in MERL database and the real-world datasets.