For a sparse symmetric matrix, there has been much attention given to algorithms for reducing the bandwidth. As far as we can see, little has been done for the unsymmetric matrix $A$, which has distinct lower and upper bandwidths $l$ and $u$. When Gaussian elimination with row interchanges is applied, the lower bandwidth is unaltered, while the upper bandwidth becomes $l+u$. With column interchanges, the upper bandwidth is unaltered, while the lower bandwidth becomes $l+u$. We therefore seek to reduce $\min (l,u)+l+u$, which we call the total bandwidth. We compare applying the reverse Cuthill-McKee algorithm to $A+A^T$, to the row graph of $A$, and to the bipartite graph of $A$. We also propose an unsymmetric variant of the reverse Cuthill-McKee algorithm. In addition, we have adapted the node-centroid and hill-climbing ideas of Lim, Rodrigues, and Xiao to the unsymmetric case. We have found that using these to refine a Cuthill-McKee-based ordering can give significant further bandwidth reductions. Numerical results for a range of practical problems are presented and comparisons made with the recent lexicographical method of Baumann, Fleischmann, and Mutzbauer.