In an earlier work with \textit{M. Ritoré} [J. Differ. Geom. 47, No. 2, 376-397 (1997; Zbl 0938.53004)] the authors studied the moduli space of minimal immersions of tori with two embedded parallel planar ends in \(\mathbb{R}^3/{\mathcal T}\), where \({\mathcal T}\) is a group of translations; they found a surface \({\mathcal S}_0\) containing the \(x_2\) and \(x_3\) axes, such that a fundamental piece consists of a disk bounded by two parallel straight lines and a segment that is orthogonal to both lines \textit{F. J. Lopez} and \textit{D. Rodriguez} [Mich. Math. J. 45, No. 3, 507-528 (1998; Zbl 0977.53006)] found a properly immersed minimal torus in \(\mathbb{R}^3/ {\mathcal S}_{(2\pi/3)}\) with two horizontal embedded ends and containing the \(x_2\) and \(x_3\) axes. This surface admits a natural uniqueness theorem, and as above it has a disk bounded by straight lines with two boundary ends as fundamental piece. In the present paper the authors generalize those results, describing the moduli space \({\mathcal M}\) of properly immersed minimal tori with two embedded horizontal planar ends in \(\mathbb{R}^3/ {\mathcal S}_0\), \(\theta\in]- \pi,\pi]\), and containing the \(x_2\) and \(x_3\) axes. Here, \({\mathcal S}_\theta\) is the group generated by the screw motion about the \(x_3\) axis of angle \(\theta\) and translation vector \(\nu= (0,0,1)\). The paper contains nicely made computer images of the surfaces.