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We investigate the complete universe evolution in the framework of $f(T)$ cosmology. We first study the requirements at the kinematic level and we introduce a simple scale factor with the necessary features. Performing a detailed analysis of the phase portrait we show that the universe begins in the infinite past from a phase where the scale factor goes to zero but the Hubble parameter goes to a constant, and its derivative to zero. Since these features resemble those of the Pseudo-Rip fate but in a reverted way, we call this initial phase as Pseudo-Bang. Then the universe evolves in a first inflationary phase, a cosmological turnaround and a bounce, after which we have a second inflationary regime with a successful exit. Subsequently we obtain the standard thermal history and the sequence of radiation, matter and late-time acceleration epochs, showing that the universe will result in an everlasting Pseudo-Rip phase. Finally, taking advantage of the fact that the field equations of $f(T)$ gravity are of second order, and therefore the corresponding autonomous dynamical system is one dimensional, we incorporate the aforementioned kinematic features and we reconstruct the specific $f(T)$ form that can dynamically generate the Pseudo-Bang cosmological scenario. Lastly, we examine the evolution of the primordial fluctuations showing that they are initially sub-horizon, and we show that the total fluid does not exhibit any singular behaviour at the phantom crossing points, while the torsional fluid experiences them as Type II singular phases.