Abstract: High dimensional tensor time series has been encountered increasingly often in applications. Factor model in a form similar to tensor Tucker decomposition has been shown to be a useful model for tensor time series. In this paper we propose a more detailed decomposition so the factors can be interpreted as global effects, main effects of individual modes (columns, rows, etc), and interaction effects among the modes. This decomposition enhances interpretability, effective dimension reduction and estimation efficiency. Theoretical investigation establishes the properties of the estimation procedure. Empirical examples are used to illustrate the applicability of the methodology, highlighting its relevance to contemporary data science challenges in high-dimensional settings.