Within the framework of 3D resistive magnetohydrodynamic, we simulate the formation of a plasma jet with the morphology, upward velocity up to 130 km s-1, and time-scale formation between 60 and 90 s after beginning of simulation, similar to those expected for type II spicules. Initial results of this simulation were published in paper by, e.g. González-Avilés et al. (2018), and present paper is devoted to the analysis of transverse displacements and rotational-type motion of the jet. Our results suggest that 3D magnetic reconnection may be responsible for the formation of the jet in paper by González-Avilés et al. (2018). In this paper, by calculating times series of the velocity components vx and vy in different points near to the jet for various heights we find transverse oscillations in agreement with spicule observations. We also obtain a time-distance plot of the temperature in a cross-cut at the plane x = 0.1 Mm and find significant transverse displacements of the jet. By analysing temperature isosurfaces of 104 K with the distribution of vx, we find that if the line-of-sight (LOS) is approximately perpendicular to the jet axis then there is both motion towards and away from the observer across the width of the jet. This red-blue shift pattern of the jet is caused by rotational motion, initially clockwise and anti-clockwise afterwards, which could be interpreted as torsional motion and may generate torsional Alfvén waves in the corona region. From a nearly vertical perspective of the jet the LOS velocity component shows a central blue-shift region surrounded by red-shifted plasma.