Many machine learning applications employ a multiclass classification stage that uses multiple binary linear classifiers as building blocks. Among these, commonly used strategies such as one-vs-one classification can require learning a large number of hyperplanes, even when the number of classes to be discriminated among is modest. Further, when the data being classified is inherently high-dimensional, the storage and computational complexity associated with the application of multiple linear classifiers can ignite critical resource management issues. This work describes a novel multiclass classification method based on efficient use of a single 'recycled' linear classifier (or ReLiC), which addresses these storage and implementation complexity issues. The proposed approach amounts to constraining the entire collection of hyperplanes to be circularly-shifted versions of each other, enabling classification procedures that may be implemented with efficient operations, such as circular convolution (which can be efficiently computed using transform domain techniques), and simple sampling/thresholding operations. We show that the optimization task associated with our proposed approach can be formulated as a quadratic program, and we introduce an efficient distributed procedure for its solution based on an alternating direction method of multipliers. Simulation results demonstrate that the performance of the proposed approach is comparable with the more complex, traditional multiclass linear classification strategies, suggesting the proposed approach is a viable alternative in large-scale data classification tasks.