We present the Spiral Classification Algorithm (SCA), a fast and accurate algorithm for classifying electrical spiral waves and their associated breakup in cardiac tissues. The classification performed by SCA is an essential component of the detection and analysis of various cardiac arrhythmic disorders, including ventricular tachycardia and fibrillation. Given a digitized frame of a propagating wave, SCA constructs a highly accurate representation of the front and the back of the wave, piecewise interpolates this representation with cubic splines, and subjects the result to an accurate curvature analysis. This analysis is more comprehensive than methods based on spiral-tip tracking, as it considers the entire wave front and back. To increase the smoothness of the resulting symbolic representation, the SCA uses weighted overlapping of adjacent segments which increases the smoothness at join points. SCA has been applied to a number of representative types of spiral waves, and, for each type, a distinct curvature evolution in time (signature) has been identified. Distinct signatures have also been identified for spiral breakup. These results represent a significant first step in automatically determining parameter ranges for which a computational cardiac-cell network accurately reproduces a particular kind of cardiac arrhythmia, such as ventricular fibrillation.