Directional dependence modeling has been applied to many research areas including economics, finance, biostatistics, and bioinformatics. The concept of directional dependence using copula regression functions has been introduced by Sungur [21]. So we propose a new copula family which incorporates the truncation invariant structure [20] into the generalized Farlie-Gumbel-Morgenstern (FGM) distributions. The directional dependence of the new truncated invariant FGM copulas will be also introduced in this research. We will show that there exists a directional dependence in our truncation invariant FGM copulas using Foreign Currency Exchange Data of the Canadian Dollar (CAD/USD), the Japanese Yen (JPY/USD), and the Korean Won (KRW/USD).