When drawing the diagonal line from top to bottom on the cylinder, not only do you not lift off with the pencil, but also the finger that is revolving the cylinder. Once you lift off with your finger the revolving stops. That took some practice (about 10 trys).
However, in practicing, I found a short cut. Start with the pencil, start at the top (on the vertical line), and then draw straight downward, and maybe to the left a bit. A spiral appears that wraps the cylinder, and with care, you can navigate that point to the bottom (right on the bottom of vertical line). This is how I achieved the image above. Then I added a mirror, and then did a copy and move.
Hope that helps. This is a good exercise.