Gene order evolution of unichromosomal genomes, for example mitochondrial genomes, has been modelled mostly by four major types of genome rearrangements: inversions, transpositions, inverse transpositions, and tandem duplication random losses. Generalizing models that include all those rearrangements while admitting computational tractability are rare. In this paper, we study such a rearrangement model, namely the inverse tandem duplication random loss (iTDRL) model, where an iTDRL duplicates and inverts a continuous segment of a gene order followed by the random loss of one of the redundant copies of each gene. The iTDRL rearrangement has currently been proposed by several authors suggesting it to be a possible mechanisms of mitochondrial gene order evolution. We initiate the algorithmic study of this new model of genome rearrangement by proving that a shortest rearrangement scenario that transforms one given gene order into another given gene order can be obtained in quasilinear time. Furthermore, we show that the length of such a scenario, i.e., the minimum number of iTDRLs in the transformation, can be computed in linear time.