We introduce the discrete asymmetric double Lindley distribution, a new two-parameter family on the integer line designed to model signed counts and net changes with flexible asymmetric tail behavior. This statistical model is obtained by merging two Lindley-type linear-geometric kernels on the negative and non-negative half-lines, with tail decay rates that are coupled through a simple two-parameter mechanism. This construction yields an analytically tractable probability mass function with an explicit normalizing constant, as well as closed-form expressions for the cumulative distribution function and one-sided tail probabilities. We further provide a transparent stochastic representation based solely on Bernoulli and geometric random variables, leading to an exact and efficient simulation algorithm that is convenient for Monte Carlo studies and validating numerical likelihood routines. Graphical illustrations highlight the role of the asymmetry parameter in controlling the imbalance between the two tails and the resulting skewness on Z. The proposed family offers a practical and interpretable alternative to existing integer-line models for asymmetric discrete data, with direct applicability to likelihood-based inference and real-world datasets.