In this paper, we present a new approach for periocular recognition based on the Symmetry Assessment by Feature Expansion (SAFE) descriptor, which encodes the presence of various symmetric curve families around image key points. We use the sclera center as single key point for feature extraction, highlighting the object-like identity properties that concentrates to this unique point of the eye. As it is demonstrated, such discriminative properties can be encoded with a reduced set of symmetric curves. Experiments are done with a database of periocular images captured with a digital camera. We test our system against reference periocular features, achieving top performance with a considerably smaller feature vector (given by the use of a single key point). All the systems tested also show a nearly steady correlation between acquisition distance and performance, and they are also able to cope well when enrolment and test images are not captured at the same distance. Fusion experiments among the available systems are also provided.