Twin support vector clustering (TWSVC) is a recently proposed powerful k-plane clustering method. It, however, is prone to outliers due to the utilization of squared L2-norm distance. Besides, TWSVC is computationally expensive, attributing to the need of solving a series of constrained quadratic programming problems (CQPPs) in learning each clustering plane. To address these problems, this brief first develops a new k-plane clustering method called L1-norm distance minimization-based robust TWSVC by using robust L1-norm distance. To achieve this objective, we propose a novel iterative algorithm. In each iteration of the algorithm, one CQPP is solved. To speed up the computation of TWSVC and simultaneously inherit the merit of robustness, we further propose Fast RTWSVC and design an effective iterative algorithm to optimize it. Only a system of linear equations needs to be computed in each iteration. These characteristics make our methods more powerful and efficient than TWSVC. We also conduct some insightful analysis on the existence of local minimum and the convergence of the proposed algorithms. Theoretical insights and effectiveness of our methods are further supported by promising experimental results.