In this paper, we introduce flexible observation-driven $\mathbb{Z}$-valued time series models constructed from mixtures of negative and non-negative components. Compared to models based on the standard Skellam distribution or on a difference of two integer-valued variables, our specification offers greater versatility. For example, it easily allows for skewness and bimodality. Furthermore, the observation of one component of the mixture makes interpretation and statistical analysis easier. We establish conditions for stationarity and mixing, and develop a mixed Poisson quasi-maximum likelihood estimator with proven asymptotic properties. A portmanteau test is proposed to diagnose residual serial dependence. The finite-sample performance of the methodology is assessed via simulation, and an empirical application on tick prices demonstrates its practical usefulness.