Communication-avoiding algorithms for Linear Algebra have become increasingly popular, in particular for distributed memory architectures. In practice, these algorithms assume that the data is already distributed in a specific way, thus making data reshuffling a key to use them. For performance reasons, a straightforward all-to-all exchange must be avoided. Here, we show that process relabeling (i.e. permuting processes in the final layout) can be used to obtain communication optimality for data reshuffling, and that it can be efficiently found by solving a Linear Assignment Problem (Maximum Weight Bipartite Perfect Matching). Based on this, we have developed a Communication-Optimal Shuffle and Transpose Algorithm (COSTA): this highly-optimised algorithm implements $A=α\cdot \operatorname{op}(B) + β\cdot A,\ \operatorname{op} \in \{\operatorname{transpose}, \operatorname{conjugate-transpose}, \operatorname{identity}\}$ on distributed systems, where $A, B$ are matrices with potentially different (distributed) layouts and $α, β$ are scalars. COSTA can take advantage of the communication-optimal process relabeling even for heterogeneous network topologies, where latency and bandwidth differ among nodes. The implementation not only outperforms the best available ScaLAPACK redistribute and transpose routines multiple times, but is also able to deal with more general matrix layouts, in particular it is not limited to block-cyclic layouts. Finally, we use COSTA to integrate a communication-optimal matrix multiplication algorithm into the CP2K quantum chemistry simulation package. This way, we show that COSTA can be used to unlock the full potential of recent Linear Algebra algorithms in applications by facilitating interoperability between algorithms with a wide range of data layouts, in addition to bringing significant redistribution speedups.