Tandem-duplication-random-loss (TDRL) is an important genome rearrangement operation studied in evolutionary biology. This paper investigates some of the formal properties of TDRL operations on the symmetric group (the space of permutations over an $ n $-set). In particular, the cardinality of `balls' of radius one in the TDRL metric, as well as the cardinality of the maximum intersection of two such balls, are determined. The corresponding problems for the so-called mirror (or palindromic) TDRL rearrangement operations are also solved. The results represent an initial step in the study of error correction and reconstruction problems in this context and are of potential interest in DNA-based data storage applications.